CORNELL UNIVERSITY LIBRARIES Ma'thematics CORNELL UNIVERSITY LIBRARY 924 1 05 225 399 The original of tliis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924105225399 SOLUTIONS OF THE EXAMPLES IN HIOHEE ALGEBEA. SOLUTIONS OF THE EXAMPLES I2T HIGHEE ALGEBEA H. S. HALL, M.A., FOBMEELT SCHOLAB OF OHKIST's COLLEGE, CAMBRIDGE, UASTEB OF THE MILITARY AND ENGINEEEINO SIDE, CLIFTON COLLEGE ; S. R KNIGHT, B.A., FOBMEELT SCHOLAR OP TRINITY COLLEGE, CAMBRIDGE, LATE ASSISTANT-MASTER AT MARLBOEOUGH COLLEGE. Uonlfon : MACMILLAN AND CO. AND NEW YOEK. 1889 [The Right of Translation is reserved.] PKINTED BY C. J. CLAY, M.A. AND SOXS, AT THE DNIVEP-ailY PBES8. ^^r m^'"' PREFACE. This work forms a Key or Companion to the Higher Algebra, and contains full solutions of nearly all the Examples. In many cases more than one solution is given, while through- out the book frequent reference is made to the text and illustrative Examples in the Algebra. The work has been undertaken at the request of many teachers who have introduced the Algebra into their classes, and for such readers it is mainly intended ; but it is hoped that, if ju- diciously used, the solutions may also be found serviceable by that large and increasing class of students who read Mathematics without the assistance of a teacher. H. S. HALL, S. R. KNIGHT. June, 1889. HIGHEE ALGEBEA. EXAMPLES. I. Pages 10—12. 8. Letr = ^=^ = ^; then c = dr, b = cr=dr', a=br=dr^; and by sub- stituting for a, h, c in terms of d, we have ^ ^^.TT ,. = r°= - ¥c + d* + b^cd^ d2" 9. Let A: = — ■' q+r-p r+p-q p+q-r then {q-r)x+(r~p)y + {p-q)z=h{[q-r){q+r-p) + ... + ...}=0. 10. _y_^y + x^x x—z z y ' _, , .. sum of numerators 2(x+u) Each ratio= s-^, ^ — - — = -i — ^''-' . smn 01 denommators x+y Thus each ratio is equal to 2 unless i + ?/ = 0. In the first case x+y X =-=2: whence x : y : z=i : 2 : 3. z y ' " In the second case, y= -x, and = - , whence a;:v:z=l: -1:0. a: — 2 2/ ,, T, ■, .. sum of numerators iix+y + z) 11. Eaohratio= — : — - — = , ^ J ^\ ' ^ 1). sum ot denominators (p + q)(a + li + c) ' Multiply the numerator and denominator of each ratio by a, 6, c respec- tively and add, then each of the given ratios _ [b + e)x+{c + a)y+{a + i)z {p+q){bc + ca+ab) *■"'■'■ From equations (1) and (2) the result follows. 12. See Example 2, Art. 12. H. A. K. », 1 2 RATIO. [CHAP. iy + 2z-x _ 'iz+2x-y _ 2x + '2y-z Multiply the numerator and ■ o 6 c I, I, denominator of each of the given ratios by - 1, 2, 2 and add ; then eacn ratio _ -(2y + 2z-a;)+2(2z + 2a!-y) + 2(2g + 2y-z) _ 5x -a + 2b + 2e 2b + 2c-a' Similarly each of the given ratios is equal to and to ; 2c + 2a-b 2a+2b-e 14, Multiplying out and transposing, b'h? + chj^ - 2bcyz + cV + a'n^ - 2eazx + aV + 5V - 2a5a:i/ = ; that is (bz -cyY + (ex - azf + (ay - bxf = 0. .•. bz-ey = 0, cx-az=0, ay-bx = 0. 15, Dividing throughout by Imn, my + nz-lx nz + lx-my _ lx+my-nz mn nl Im _(nz + lx-'my) + {lx+my-nz} _ 2lx _ x ~ nl+lm '~ nl + lm m + n' Thus we have x _ y _ "■ y+z-x y + z-x m+n~ n+l~ l + m" (n + l) + (l + m)-(m+n) ~ 21 Hence the result. 16, From ax+cy + hz=0, cx+by + az=0, we have by cross multiplica- tion, "^ — y — g _7. Ti^^^ ~ U^-' ~ ab^^~ ' ^^'^• Substituting in the third equation bx + ay + cz=(i, we have l(ac-W) + a(bc-d?) + c(db-c^) = Q. 17, From the first two equations we have by cross multiplication, as _ 3/ _ z hf-bg gh-af~ ab-h?' Substituting in the third equation, we get g(M-''>9)+f{ah-af) + c(ab-W)=Q. 18. From the first and second equations, X _ y __£_ ac + b bc + a 1-c' From the first and third equations, X _ y _ z ab + c~ l-b^~ bc + a .(1). .(2). I-] BATIO. From (1) and (2) y y. y " 6c + o^ 1-62 l-c" z Xl 1 bc + a or 3,8 2= l-62~l-c=' 19. From the first two equations, X _ 9 t oi> + a ab + b 1-ab Substitute in the third equation, /. c{ab + a] + c{ab + h)=l-uh. 22. From the first and second equations we have by cross multiplication, yz zx xy ., , . X y , y 23. From the first and second equations by cross multiplication, a^ y^ z^ 45^80^5' "^^^^^^ a;=±3z; y=J=iz. 24. From the given equations by cross multiplication we obtain the ratios of Z : m : li. The value of I is proportional to Ub - ^c) (Jc + J a) (V* + s/c) yc -^a)' that is, proportional to 7^ — ^ r , ^ (6 - c) (c - o) so that we may put 1= = — M r h. •^ (6 - c) (c - a) XT 2 ^ Hence {a-b){c-^ab) {b-c){e-a){a-b)' By symmetry we obtain the required result. 25. From the first two equations, « _ y z _. o(63_c2) -b{<~'-a') '€{0^- b-') ~ ^^' substituting in the third equation, we find that k-=l. See Example 3, Art. 16. 26. From the first two equations, ~ = k say; 6c (6 - c) ca (c - a) ab (0-6) and from the third equation we find that £= 1. 1—2 4 PROPORTION. [chap. 27. From the first two equations, X ^ y _ z (1). db + a ab + b 1-ab From the second and third equations, X _ y ^ z (2). 1 - 6c bc + b bc + c From (1) and (2) ,f' ,, x .j-^ = .p^ x -yy^, ; * ' ^ ' a (6 + 1) 1-bc 1-ab 0(6 + 1) . a^ _ z^ a(l-bc)~ c(l-ab)' 28. From the first and second equations, ^ = y - ^ (1). hf- bg gh -af ab- W From the second and third equations, ^ ^ y ^ ^ (2). ftc-/" fg-oh hf-bg ^' From the first and third equations, _^_= y ^-J— (3). fg-ch ca-g^ gh-af From (2) and (3) ^^ x -^ =^5^^ x ^^ ; x^ _ y^ bc-f'-'~ ca-g^' From (1), (2), and (3) X y z _ y z X be -/^ ' ca-g^' ab-b? fg- eh ' gh-af hf-bg' hy equating the denominators the second result follows. EXAMPLES, n. Pages 19, 20. Examples 4, 5, 6, 7 may all be solved in a similar manner; thus take Example 6, and put ^ = -5 = 7c, so that a=bk, c=dk; then a-c _ bTc-dlc _ Tcjj^ + d^ ^ Jk'b-^ + Tt'd' _ Ja' + e'' b-d b-d Jb'^ + d'' sjb^ + d^ ^/P+T^' Examples 8, 9, 10 may all be solved in a similar manner. Put - = - = - = fc, so that c=dk, b = dW, a = dTt^, then in Example 9, 2a + 3d_2dP + Sd _ 2k^ + 3 _ 2 63^ + 36' _ 2a^ + 3h^ 3a - 4d ~ 3dk^ - 4d ~ SA* - 4 ~ Sft'/jS - 46^ ~ 3a^ - 46^ " II.] PROPORTION. 11. Put j = - = k;ttiena = bk,c^r, be k j^+^__H!Lllil. and . "■ ~" •^'- - = ^ "/ _ 54 """-'■■"■■ .Kp-'-") 13. Componendo and dividendo, i— =s i-r . whence a; = 0, or x + 1 5s -13 ' 2a:-3 _ 3x-l x + 1 5a; -13' 15. Componendo and diyidendo, :;^_ww — 5. j^^ clearing of frac- 71X "1" fix "T" C tions we obtain a simple equation. 16. We have a(a~b-c + d) = a^-ab-ac + ad=a'-ah-ac + hc = {a-b){a-c);. (a-b){a-c) a- b-c+d=- a 18. The work done by a; - 1 men in x + 1 days is proportional to / ■.^/ 1^ 1, (a;-l)(x + l) 9 (a;-l)(x+l); hence J^ = ^. 19. Denote the proportionals by x, y, 19-2/, 21 -x. Then x(21-x)=2/(19-2/) (1), and x« + 3/2 + (19-^)2 + (21-x)2=442 (2). From (1) x2 - 2/2 - 21x + 191/ = 0. From (2) a:°+2/2-21x-19i/ + 180 = 0. Add x2-21x + 90=0, x=6 or 15. Subtract y^-19y + Q0 = 0, y = 9 or 10. 20. Let the quantities taken from A and B be x and y gallons respec- tively. Then 21. Suppose the cask contains x gallons; after the first drawing there are x-9 gallons of wine and 9 gallons of water. At the second drawing X — 9 — — X 9 gallons of wine are taken, and therefore the quantity of wine left is X 9fx — 9) (x — 9)2 (x - 91 ^ = . Hence the quantity of water in the cask is 6 PEOPOETION. [CHAP. II. X X :. (x-^Y :18x-81 = 16 :9, or (a; -9)2 =16 (2a; -9). 22. Denote the quantities by a, ar, a'fi, wfi. Difference between first and last _ ar'~a DifEerence between tbo other two ~ or* ~ r r^+r + 1 3r+(l-r)2 „ , (1-^)' . = = — — o+ — - 1 r r r and this is greater than 3. 23. Let T amd C denote the town and country populations ; 18 the increase in the town population is ^j^ ^> 4 „ country Tnn^'^ 100 total m^^^^' .: 18r+4c=15-9(T+C). 24. Let 5x and x denote the amounts of tea and coffee respectively. On the first supposition, the increase of tea is r^ x 5a; ; the increase J . 7c of coffee is -ryrj; X X ; and the total increase is r^ x 6a;. .-. 5o + 5=42c (1). On the second supposition, we have 5S + a = 18c (2). 5a + 6 7 From (1) and (2), 56 + a 3 25. Suppose that in 100 parts of bronze there are x parts of copper and 100 - a; of zinc ; also suppose that in the fused mass there are 100a parts of brass, and 1006 parts of bronze. 100a parts of brass contain ax parts of copper and a(lOO-x) parts of zinc. Also 1006 parts of bronze contain 806 parts of copper, 45 parts of zinc, and 165 parts of tin. Hence in the fused mass there are aa; + 805 parts of copper; a (100 -a;) + 45 parts of zino, and 166 parts of tin. a.^ + 806 _ g (100 - a ) + 46 _ 165 74 ~ 16 " 10 ■ 111.] VARIATION. .-. 10 (oa; + 805) = 74x166; that ia 10aj;= 3846. Also 10 {o (100 -a;) + 44}=16xl66; that is 10a (100 -x) = 216 J. lOflx 3846 X 16 , or: •■ lOa(lOO-x) 2166' lOO-i" 9 ■ 26. Let X be the rate of rowing in stiU water, y the rate of the stream, and a the length of the course. Then the times taken to row the course against the stream, in still water, and with the stream are . - , minutes respectively. x-y X x+y "^ •' Thus -^=84 (1), x-y ^ " ^--^ = 9 (2). X x + y ^ ' From (1), a=8i{x-y), . S4(a;-y) U(x-y) ^^ X x + y ' .: 2Sy{x-y)=3x[x+y), or 3i' - 25w/ + 28i/» = 0, x = 7y, or Sx=iy. If x=7y, then a=84x6«, and time down stream=2-=63 minutes. Sy Similarly in the other case. EXAMPLES, m. Pages 26, 27. 7. P=-^ . where m is constant; hence j = m x = x -=- ; thus f» = l, and Q=PJ?=,y48 x ^75 =60. 9. Here y = mx + - , where to and n are constants ; whence 6 = 4m + - . and -=- = 3m + 5 . From these equations we find m = 2, n = - 8. o o 10. Here y=mx+ -^, so that 19=2m + j, and 19 = 3m + -^ ; whence m=5, n = 36. 8 VARIATION. [CHAP. 11. ^=^^, BO that 3=^^^^, and therefore m = I ; hence 12. Here x + y=m[z+-\ , and x-y=n I z — |. From the numerical data, 4=mx24, and 2=raxlJ; thus x + y = -^(z + - 1 and x-2/ = -(z — j. 44 4 By addition, 2x=j^z + yf' • 14. Here y=m + nx+px-. from the numerical data, 0=m + n+p; l=m+27i + 4^; 4=m + 3?i + 92); whence m=l, m=-2, p = l; and y=l-2x + x^={x~iyK 15. Let s denote the distance in feet, t the time in seconds : then s oc «', so that s=mt^. Now 402J=m x 5', hence m=16-l. In 10 seconds, s = 16-l x 102=1610. In 9 seconds, s = 16-1 X 92=1304-1. The difference gives the distance fallen through in the ID"" second. 16. Let r denote the radius in feet, V the volume in cubic feet; then Focr^ so that V=mr^. Hence 179| = mx(|j ; 7 n\^ 1 /TX** 1 when r= J , r=m x ^ jj = g x "» x (^^ j = 8 x 179J=22H. 17. Let w denote the weight of the disc, r the radius and t the thickness ; then w varies jointly as r' and t; hence w=mtr^. If w', r', t' denote cor- responding quantities for a second disc, w'=mt'r'^. „ w tr^ Hence — , = -r-r-, . w tr^ t 9 w 9^2 If -> = 3 and — =2, we have 2= — y^ , that is 3r=4r'. r 8 w 8/2 18. Suppose that the regatta lasted a days and that the days in question were the (^-l)'^ x"", and [x + 1 )"*. Then the number of races on the a;"' day varies as the product x(a- x-1). Similarly the numbers of races on the (s-l)"' and (a; + l)"' days are proportional to (a; - 1) (a - a; - 2). and (x + \){a~x). in.J VARIATION. 9 Hence {x-l){a- x-2)=6k (1), x{a- x-l)=5k (2), {x + l){a-x) = 3k (3). Subtract (2) from (1), 2x-2-a = lc (4). Subtract (3) from (2), 2a;-a = 2A: (5). Subtract (4) from (5), 2 = k. Hence from (5), 2x-a = 4; that is o = 2a;-4 (6). Also from (2), a;(a-x + l) = 10, and substituting from (6) , a; (a: - 3) = 10. Thus a;=5anda = 6. 19. Let £p be the cost of workmanship ; w carats the weight of the ring; £x the cost of a diamond of one carat; £y the value of a carat of gold. Thus a=p + {w-3)y + 9x, b=p + {w~i)y + l&x, c—p + {w-5]y + 2Sx, .'. a + c-2b = 2x, whence x=—- J. 20. Let £P denote the value of the pension, Y the number of years; then by the question P oo ^/Y, that is P=m^Y (1). Also P + 50=mjY+y (2), and ' JY 8 ' ^" From this last equation r=16, and therefore from (1) and (2), P=im, P + 50 = 5m. 21. Let F denote the force of attraction, T the time of revolution; then i^'xg.andT^xJ. M Tl-iusDocJ'TS; that is D x -^ T^^ oi MT^xD'; :. MT^=kDK fmf 2 (7 3 Thus 7n^t^=hd^, and m^t^^kd^^; that is — ^= -j^- TflnZn tin d 35 Using the numerical data, ^^ = oT • ^=343, and f2= 27-32 days. m, SiSti^ 35x35x35 •'• {27-32)2 ~ 31 X 31 X 31 ' 10 ARITHMETICAL PEOGRESSION. [CHAP. 27-32 X 5 / 5 13-66 , __ ,^^ ••• '-^r- '^ V 3i=3r72^=''^' ^^^'- 22. Let re be tlie rate of the train ia miles per hour, g the quantity of fuel used per hour, estimated in tons; then q = kx^; but 2=ix(16)i; 2 2 ••• 2 = 256^ = .•. the cost of the fuel per hour is £ „ x ofifl^'" * 256 ' \ x^ X .-. cost of fuel per mile is £- x 256~^256 ' Also cost for journey of one mUe. due to " other expenses," is * 20 k" 16a;' .-. cost of journey per mUe ^^ £ ( ggg + jg^ ) > and this has to be as small as possible. Now this expression=/'^- T-T-j + KTy and therefore is least when 'Jl--1^=Q; thatis, x = 12. lO i^JX Hence the least cost of the journey per mile is f/^, and the cost for 100 miles is £^^=£9. 7s. 6d. EXAMPLES. IV. a. Pages 31, 32. 18. « = 5 (a + J) ; thus 155 = 5 (2 + 29), and n= 10. Again Z=o + (m-l)(J, that is, 29=2 + 9(?. 20. Here 18 = a + 2(i, 30=o + 6i, so that a=12, d=3. 21. Denote the numbers by ^ - li, a, a+d, then 3a=27, thatis, o = 9. Hence (9-d)x9x (9 + d) = 504. 22. The middle number is clearly 4, so that the three numbers are i-d, 4, 4+d. Thus (4-d)3 + (4)3 + (4 + d)3=408. IV.] ARITHMETICAL PROGRESSION. 11 23. Put n = 1 ; then the first term = 5 ; put n=15; then the last term =61. Sum=— (first term + last term) = -^x66=495. Example 24 may be solved in the same -way. 25. Putn=l; then the first term = - + &: a put 78=2); then the last term = - + 6; a :. sum=| (first term + last tenii)=| f^^ +2b) . 26. The series = 2a- i, ia--, 6a--, a a a .: /S=(2a + 4a + 6a + ... tore terms) /I 3 5 X . \ -l-H 1 h ... to n terms . \a a a J EXAMPLES. IV. b. Pages 35, 36. 3. Hereo + 2d=4a, and a + 5d=17; henee a=2, d=3. 31 1 13 4. Herea + d=— , a\?,M=-^, a + (re-l)(i=-— -; so that d=- J , a = 8, ra=59. 6. Denote the instalments by o, a + d, a + 2ci ; then sum of 40 terms = 3600; o and sum of 30 terms = 5 of 3600=2400. o .-. 20(2a + 39d) = 3600, and 15(2a + 2M) = 2400; .-. 2a + 39d=180, 2a + 29(Z = 160. 7. Denote the numbers by a and I, and the number of means by 2m. Then a + Z=-5-, and the sum of the means = 2mx -jr— =m(a+Z). But this sum=2m+l; 13 .". 2m + l = m(a + Z)=-^m, whence m = 6; and the number of means is 12. 12 ARITHMETICAL PROGRESSION. [CHAP. 9. The series is Iz^ . _5_- Lt^^ and is therefore an A. P. whose 1-x 1-x !-« first term is , ^ and difference :P^ — . 1—x l~x Hence S= ^ {^j^ + ^^f = ^p^^ {2 + (» - 3) V-J. 10. We have | {2a + 6(J} =49, that is a + 3d=7. Similarly ^{2a + 16d} = 289, that is a + Sd=n. Thusa=l, d=2. 11. Let ic be the first term, y the common difference ; then a=x + {p-l)y, h = x + (q-\)y, c=x + {r-l)y; :. (ti-r)a + (r-j))b + ('ja-q)c = 0, since the coeflSoients of x and y will both be found to vanish. 12. Here | {2ffi + (^-l)d} =2; that is, 2a + (2)-l)d=?2. Similarly 2a + (j-l)(?= — . Whence d=-2fi + lV a=^ + ^-i-- + l. \P qj i P P q _._^^^|2p^2,_2_2 /I IM 13. Assume for the integers a -3d, a-d, a + d, a + 3d; the sum of these is 4a; thus 4(i=24 and a =6. .-. (6-3d)(6-d){6 + d)(6 + 3d) = 945, that is, 9 (2 - d) (2 + d) (6 - d) (6 + d!) = 945. 14. Assume for the integers o-3d, a-d, a + d, a + Sd; thus from the first part of the question a=5; and from the second (5-3d)(5 + 3d) 2 ^ (5-d)(5 + d) =3; whence d=l. 15. Here a + (p-l)d=j, and a + (g-l)d=ji; whence d= - 1, a=p + q-l. Thus the TO* term=^ + j-I + (TO-l)(-l)=^4.2_m. 17. Putting n=r, the sum of r terms is 2r + 3r^; putting n=r-l. the sum of {r - 1) terms is 2 (»• - 1) + 3 (»■ - 1)2. The difference gives the ?•* term. IV.] ARITHMETICAL PROGRESSION. 13 18. We have m{2a+m^l .d) ^r^ n{2a+n-l.d) «" thatis, ra(2a+m-l .d)=m{2a+n-l . d); whence 2a=d. rpjj m'''term _ a + (TO-l)d _l + 2(m-l) 2m-l n»i> term ~ a + (n-l)d~ 1 + 2 (re -1) ~ 2re-l ' 19. Let m be the middle term, d the common difference, and 2^ + 1 the number of terms ; then the pairs of terms equidistant from the middle ;tcrm are m-d, m + d; m-2d, m + 2d; m-3d, m, + 3d; m-{p-l)d, m + {p-l)d. Thus the result follows at once. 20. See the solution of Example 17 above. 21. Let the number of terms be 2re. Denote the series by a, a + d, a + 2d, a + 3d, a + {2n-l)d. Then we have the equations : |{2a+(re-l)2d} = 24 (1), ~{2{a + d) + {n-l)2d} = 30 (2), (2n-l)i=10i (3). From (1) and (2), nd=e (4). From (3) and (4) re =4, and the number of terms is 8. 22. In each set the middle term is 5 [Art. 46i Ex. 1]. Denote the first set of numbers by 5-d, 5, 5 + d; then the second set will be denoted by 5 - (d - 1), 5, 5 + (d - 1) ; hence {5-d){5 + d) _7 _ (6-d)(4 + (iJ~8' whence d;=2 or -16. The latter value Is rejected. 23. In the first case the common difference is — — 5- : and the r"" mean, n + 1 T (2y — x) being the (r+ 1)"' term, is x+ .■, • r{y~2x) In the second case the r"" mean is 2x + ra+1 m+1 re+1 (n + l)a! + r(2i/-a:) = 2(ra + l)a; + r(y-2a;), .-. ry = {n + X-r)x. 14 GEOMETBICAX PEOGEESSION. [CHAP. 24. Here ^{2a + {p-l)d} =^{2a + {q-l)d}, .: {2a-d)p+p^d={2a-d)q + qH; {2a-d){p-q) + {p^-q')d=0. 2a-d + (p + q)d=0, or 2a + {p + q-l)d=Q. .-. ^{2(i + (i) + 2-l)4=0; that is, tlie sum otp + q terms is zero. EXAMPLES. V. a. Pages 41, 42. „„ o(r8-l) 9a(r3-l) „ , „ 20. -Tnr- r-1 ' ••■r°+l=9;'-=2. 21. ar*=81, or=24; .-. j-=| and o=16. 22. 23. Use the formula s = ^-^. ' r-1 24, 25. The solutions of these two questions are very similar. In Ex. 25, assume for the three numbers -, a, ar; then -xasxar=216: wnence r' ' ' r ' ft a = 6, and the numbers are - , 6, 6r. T Again, (^x6^ + (6x6r) + ^^ x6r"j = 156; 3 1 that is, — i-3r=:10, whence r=3 or ^, T o 27. Let/ denote the first term, x the common ratio ; then a=fxP-\ h=fxi-'; c^fx'-K .: oS-'"6'"-Pc»-«=/«-'^T-m'-«a;a>-i)w-'-m8-i)o--3))+(r-i)(ji-«)=fOa.o= L 28. Here ^±-=4, and j^=192. from the first equation o = 4 (1 - r) ; hence 64(l-r)' ,„„ 1-ya =192, or (l-r)==3(l + r+r=), V.J GEOMETRICAL PROGRESSION. 15 that is, 2r='-5r+2=0, whence r=2 or ^ . The first of these values is inadmissible in an infinite geometrical progression ; the other value gives a =2. EXAMPLES. V. b. Pages 45, 46. 1. S=l + 2a+Sw'+ +na^^-\ aS= a + 2a«+ + (n-l)a"-i + na"i .-. S{l-a) = l + a+a^+ a'^'^-na'^ =- na". 1-a 3 7 15 . 31 ^=l + 5+i6 + 6i + 256+- 4 4^16 64^256 3 „ , 2 4 8 ,16 , By subtraction, 4 ^=1 + 4 +16+64''' 256'^ ,1111, o = 1 + 2 + 4 + 8 + 16+ =2- 3_ S=l + Sx+5x^ + 73i? + 9x*+ .•. xS= x + dx^ + 5x^ + 7x^+ By subtraction, {l-x)S=l + 2x + 2x^ + 2x^ + ix* + _ 2x _! + « "" l-a;~l-a!' 2 3 4 n 4. S = l + 2 + 25'^25 + ■''2^^' 1 12 3 n-1 n "2^= 2'^¥'^2^'^ + 2i=r + 2''' 1 11 1 ^ By subtraction, 2^~''''^2'''P''' + 2^^'" 2" 2" iL_Q_A ii — 1 2"" 2"" 2™' ■"•■"2 5. ^=1+1+1 + 1+ . is- i+?+5+ "2 2+4+8^ By subtraction, 2^^^'^^'^2^i''' = 1+2=3- 16 GEOMETRICAL PROGRESSION. [CHAP. 6. S = l + 3x + 6x^+10x^+ /. xS= x + 3x'+ 6a;'+ By subtraction, (l-x)S = l + 2x + 3x'+ ia^+ 7. Jjetp and q be the common ratios of the two progressions; then b = ap^, and 6=iig*; hence ^ = 2^ ■ that is, (n+ 1)'" term of first series = (2?i+l)"' term of second series. 8. The sums are "^ ~ ' and ''I ~ ' respectively; and since these r - 1 r^-1 are equal 7 = -s — r; •'• o=a(r + l) = a + ar. r -1 r^ — 1 ^ 9. S=l + (l + S)r)-(l+6 + 62)r2+(l + 6 + 62+J5))-3 + .-. rS= r-l-(l + S)j-2 + (l+,6 + 62)r3 + By subtraction, {l~r)S=l + ir + iV + b^r^+ = irbr' 10. Wehave a + ar+ar'' = 70 (1); 4a + 4ar''=10ar (2); from (2), r=2 or -. 11. We shall first shew that the sum of an infinite G.P. commencing at any term, say the {n + iy\ is equal to the preceding term multiplied by l — r 1-r 1-r r 1 1 In this particular example, the value of = is -, so that r=j. Again a+ar=5, hence a=4. 12. S=(x + x^ + x^+...) + {a+2a+3a+...); the first series is in G.P., the second in A. P. 13. S={x^ + x* + x^+...) + lxy+xh/^+ a?y^+...); here both series are in G.P. 14. S = (a + 3a + Sa+ ...) + ^---+ — -... J ; the first series is in A. P., the second in G.P. 15. The series may be expressed as the sum of two infinite series in G.P. 16. The series may be expressed as the difference of two infinite series mG.P. ^■] GEOMETRICAL PROGRESSION. 17 17. B.eT6 - = - = -; hence b'' = ao, c^=bd, ad=bc. Thus {b-c)^+{c-a)''+{d-bf=V-2bc + c^ + c''-2ca + a^ + d'-2bd + b^ =a^-2bc + d^ = a^-2ad + d''={a-d)\ 18. Here ^~- = 2 J^; so that {a + bf = 16ab, or a' - liab + 62=0; that is, ^|y_ 14^1^+1=0. 19. Giving to r the values 1, 2, 3,...n, we have S=3.2 + 5.22 + 7.2'+ + (2ra + l)2»; .-. 2Sf= 3.22 + 5.23 + + {2n-l)2''+(2re + l)2"+i. Subtracting the upper line from the lower, S=(2re + l)2'»+i-3.2-(2.22 + 2.2S + + 2.2") = (2ra + 1) 2"+! - 6 - ^ ^^"~ Y ^^ = (2ra+ 1) 2"+i - 6 - 2 . 2^+1 + 8 = M . 2"+2 - 2"+! + 2. 20. The series ial + a + ac + a^e + aV + a^e" + to 2n terms = (l + ac + aV+ ... ton terms) + a (1 + ac + a^cS + to n terms) = {l + a){l + ac + aV+ to m terms) _( l+a)(a"c''-l) ~ ac-1 a Ir^ — 1) 21. We have S„= ' _.. — , and by putting in succeEsionM=l, 3, 5,... we obtain the values of Sj, Sj, Sj, ... Thus the required sum =-Ar \{r-l) + (r'^-'i.) + {r^-l) + ... to m termsj- = — - ir+r'+r^+... to re terms -ml r-l [ j 22. We have 5'i=-i- = 2; S., = -^=3; 1-- 1- 2 3 'S3=-^-4, &0.; :Sj,= ^ =p + l. 1-4 1- 4 i) + l 2^ sum = 2 + 3 + 4+... topterms=|{4 + (i)-l)}=|(p+3)". ' H. A. K. 18 THE PROGRESSIONS. [CHAP. 23. Wehave l+r+r^+r^ + +i-°"= -^_y • Now (l-r™)" is positive; that is, 1 - 2r^+r^'^>0, or l + r2»>2)». Sinularly r(l-r'»-')2>0; that is, r- 2r™+r«"'-'>0 or r+r^"" i>2r'»; and generaUy r^ (1 + 1-™-*)^ > 0, that is r^ - 2r"' + r"'^-" > 0, that is rP+r2™-P>2r™. Now i+r + ?-=+rS+r"'+ +r*" = (l + r2'») + (r+r2"-i) + (j^ + '-="~^) + +'^. and is therefore greater than .2r'" + 2r™ + + r™, that is greater than (2OT+l)r™. .-. (2m + 1) r" < \rl^ , that is (2m + 1) r™ (1 - r) < 1 - ?-2^+^ Multiply both sides by j-™+', thus (2m + 1) j-^""*! (1 - r) < r"+Ml - J'™"^^)- w+l Put 2m + l=n, then Br''(l-r)'i term, and S3-S2=6 + 5c = 3'*term; /. the first three terms are a+i + c, h + Zc, h + 5c; .: after the first term the series is an A. P. whose common diff. is 2c. Also the re"' term = 6 + (2 re - 1) c. 19. The re'" term=4re3 _ Sn? + 4re - 1, .-. S=4Sji3- 6Sre2 + 4Sre- K=re*, after reduction. 20. Let X, y be the two quantities, then y-A^^A^-x, or x + y = Ai + A2 (1), ^^ = ^, or xi/ = Gi(?2 (2), y H, J?i x'" xy H^H^ ^^■'• Divide (1) by (2) and equate to (3). 21. We have p= =-, g = — ^ — t-^; by eliminating b we get the equation na''-a{(n + l)p + {n-l) q}+npq = 0. For real roots we must have {{n + l)p + {n-l)q}'-^n'pq positive; that is, (ra + 1)^2)= - 2pq {n" + 1) + (re - 1)= q^ or {(™ + l)''i' — (i-lj^sKp-?) must be positive: .•. q cannot lie between p and ( z ] p. 22. S=S(a + re-l.d)3 , 3a2d(re-l)re 3(i(«2(re-l) re(2re-l) d^{n-lW =na? + L__i— + ! 2—^ -' + ' ^ — 2 6 4 = I {4a3 + HaH (re - 1) + 2ad'^ (re - 1) (2re - 1) + d% (re - 1)=} = |(2a + re^. d) [ia^ + i (n-1) ad + n(n-l) cP} = g(2a+ ^^. d} !«='+ (™- 1) 0^+ ^i^ dj , M(re-1) . . , which proves the proposition, smce —5-= — - is an integer. 22 PILES OF SHOT. [CHAP. EXAMPLES. VI. b. Page 56. 4. Place on the given pile a triangular pile having 13 shot in each side of the base ; then . , . , , . M 25.26.27 No. of shot m the complete pue= ^ , No. of shot in the added pile= — ^-77— — • .-. reauired number= 3x^(50x9-14x5} = 2470. 5. The required number is — '—^ '-^ — , which reduces to 21321. 6. We have to find m from the equation 34.35(3m-33) ^^3^gg^ 6 17. 35 (m- 11) = 23495, whence m=52, 7. The no. of shot in a complete pUe which has 33 in a side of the tase is ^^ ^ ^^ ^ ^'^ , or 11 x 17 x 67, that is 12529. b 12 X 13 X 25 In a pile which has 12 shot in each side of the base there are ^ . or 650 shot; .-. the required number =12529 -650 = 11879. 8. Since there are 15 courses, and the pile is complete, re = 15, and m = 20 ; .•. by the formula of Art. 73, the number is 1840. 9. Add a rectangular pile having 10 and 17 shot in the sides of its base, 10 X 11 X 42 then the no. of shot in this pile is ^ , or 770. Also there are 20 D courses, so that the base of the complete pile has 30 and 37 shot in its sides ; /. no. of shot in the complete pile = x = 12710: .•. no. in the incomplete pile =11940. 5 X 6 X 33 10. By formula of Art. 73 the required number is ^ , or 190. 11. Let n be the no. of layers, then, by Arts. 71 and 72, we have ji(re+l)(re + 2) n(re+lH2m + l) ,„ 6 12 -^™' VI.] PILES OF SHOT. 23 and we have to find the value of — ^5 — - . XT n{n + l) [71 + 2 2ra + l) ,„ Now -^i-3 ^1=120' whence 'iM = 300. 12. Let n be the number of shot in a side of the base, then we have ^2- (to -15)"= 1005, whence n=41. We have now to find the number of shot in an incomplete square pile of 16 courses when there are 41 shot in a side of the base. This is 41 . 42 ■ 83 25 . 26 ■ 51 6 6 which reduces to 18296. 13. We have to shew that w(?t+l)(2re + l) _l 2K(2?i + l)(2n + 2) 6 ~4" 6 ■,„ vn- 1, n{n+l){n + 2) 13 14- ^^^"^^ 2n(2n+l)(in+ir WS' ■whence lire" - 123ra - 108 = 0, or (lire + 9) (™ - 12) = ; whence n = 12. Thus the number of shot in triangular pile = — '—^ — = 3G4; „ , i . •, 24.25.49 ..„„ and the number of shot in square pile = ^ =4900. 15. The no. of shot in the pUe = x -jg- = 680, 6 re(re + l)(re + 2) = 6x 17x40 = 15x16x17; 1- . re(n + l) i„„ whence n=15, .-. -^ — = 120. , , . . ., re(re + l)(2re + l) 16. The number of shot m square pile =—^ ^ . . , . • i • , -, n(n+l)(n + 2) The number of shot in triangular pile= —5- '^ . The difierence=^^i^(2» + l-» - 2) = ( "-1)^("+1) , and this is the number of shot in a triangular pile which has re - 1 shot in a side of the 24 SCALES OF NOTATION. [chap. EXAMPLES. VII. a. Page 59. 1. 23241 % 303478 3. 3673124 4032 150732 1732765 300421 264305 1740137 333244 728626 4. 3ie756 5. 1131315 6. 6431 2fi46«2 235143 35 e7074 4 ) 452132 112022 45115 25623 334345 7. 4685 £ . 36 ) 102432 ( 1625 9. 11022201 3483 36 121012 15276 334 1201)10201112(2012 42154 321 10102 21072 133 2211 15276 105 1201 17832126 252 252 10102 10102 in 360il4 342 11. tm 114 14 tm 1232 1101 1021 9tttl 9tttl 3014 9tttl 3014 9t«tl tt«90001 12. 2541 231 3102 2641 231 231 231 Or thus: mt= 10000-1, and (10000 1)^ = 100000000 + 1 - 20000 = ««90001. 13. 6541)14332216(1456 6541 44612 36124 G. u. M. = 231. 14. 103050301 20404020 62444261 ( 7071 61 15. 64551 45665 65536 55536 eeJOOi ( eee tl 1607 16161 14442 14261 Ite lete 16161 16161 19e01 19e01 VII.] SCALES 6^ NOTATION. 25 16. 2 3102 31141 10; 3102 242 242 242 31141 3102 121 .•. the G. 0, M. = 121,' In the scale of six we have 23=3x5, 24=4x4, 30 = 3x3x2, 32 = 4x5, 40 = 2x3x4, 41 = 5x5, 43 = 3x3x3, 50 = 3x5x2; .-. the L. 0. M. = 33 X 52 X 42= 122000. 7 ) 4954 7) 707.. .5 7)101...O 7) 14... 3 2...0 2 )206 2)103...Q 2)51.. .1 2)25.. .1 2)12. ..1 ■2)J...O 2^...0 1...1 9 ) 5381 9) 597.. .8 3 )66...3 7.. .3 6)6*12 e)J56...S e)_81...7 8.. .9 VII. b. Pages 65, 66. 2. 5)624 5) 124.. .4 5) 24.. .4 4.. .4 4. 3 ) 1458 3)486. ..0 3)162...0 3)54...0 3)18,..0 ;3)6...0 2...0 5 ) 212231 7. 5 ) 13233. .2 5 ) 1203. .0 5 ) 103. .4 3. .4 9 ) 213014 10. 9)13001. .1 9 ) 1000. .1 9)40. .0 2. .6 t ) 398e t)46*...7 t)55...8 "He. ..5 8)23861 8 ) 2663 .. .4 8)307...! 8)34.. .2 3.. .7 26 11. 13. SCALES OF NOTATION. 5)400803 12. T) 20665152 [chap. 5) 71872.. .2 T ) 1161414. ..3 5) 13885... 4 T) 50500., ..« 5) 2534.. .3 2')2655, ..« 5) 460.. .4 2')151.. .0 5) 88.. .3 r)io.. .1 5)16.. .0 0.. .7 a...o tUeee or t ) ttteee T f) 111124.. .7 130 t) 13862.. .8 T 1570 i)16«2...6 t)in..A T 18851 T t)23...1 2...7 226223 T 2714687 3x7_„, . 1x7 7x7 ., 9x7 „, ^(T" rn. -io-=Ofir; -^=4A; -T7r=6iV; after this the figures recur. 14. 10 10 15. 17 in scale ten=15 in scale twelve. •15625 1-875 T fo T 6 Or thus : •15625=i 5x12 7xl2_ 1x12 32 ~^' 8 -'+*' -—2~' 16. 9 )200 2...0 •211 9 7-1 _9 3 17. 8 )71 8)_t...5 1...2 •03 8 0-2 8 1-4 _8 2-8 _8 5-4 after this the figures reonr. TII.J SCALES OF NOTATION. 27 18, The septenary numbers 1552 and 2626 are equal to the denary numbers 625 and 1000 respectively; and --— - = 5 . 1000 8 X9. •1..4«.....|4+*+...=l^(l-l).J.|. ■'*~-=(?4)-(J4)-(^f.)-- _/4 SN/ ]^\_; 30_5 '48~8' 20. If »■ be the radix of the scale, then 182 = 2)-2 + 2r+2; that is r2 + y_90=0, or r=9. 21. Let r denote the radix of the scale, then ^ = T2 + ^ ; that is 25r* - SSir? _ 256 = 0, or (25r2 + 16)(r2-16) = 0; thus r=4. 22. Here 5r2 + 5r+4=(2r+4)2; that is, r^ - llr - 12 = 0, or r = 12. 23. The second number appears the greater, and therefore its radix is less than ten ; also the radix must be greater than 7 ; thus the radix is either 8 or 9 ; and by trial we find that it is 8. (4r?+7r + 9) + (9r2 + 7) = 2(6r3 + 9r+8); r''' — 11?'=0, or r=ll. 24. Here tbat is. 25. Here that is [r+r^j (r+r2)-(r) ' r r" ' or »-'-10r-24=0, and r=12. 26. The second number appears the smaller; hence the radix must be greater than 6 ; also it must be greater than 8 ; hence it must be one of the numbers 9, 10 ; by trial we find that it is 10. 8 4 2 27. r'' + 4r+8H (--5 is the square of r + 2+-. 28. r5 + 2r5+3r* + 4r3 + 3r2 + 2r+l is the square of j-^ + r^ + r + l. 3 3 1 1 29. 1+- + T + -3 is the cube of 1 + -. 28 SCALES OP NOTATION. [CHAP. 30. , One ton = 2240 lbs., and we have to express 2240 2 ) 2240 in the binary scale. 2) 1120... Thus, 2240 = 2"+ 2' + i 2)560...0 2)280...0 2)140...0 2)70...0 2J_b5...0 2)17...1 2)8.. .1 2)j4...0 2)^...0 1...0 31. We proceed as in the last Example, and 3 ) 10000 express 10000 in the scale of three. In dividing 41 g \ 3333 i by 3 we have a (Quotient 13 and remainder 2; since, si mi n however, only one weight of each kind is to be used ; 1111...U we put 14 as the quotient and - 1 as the remainder, 3 ) 370 . ..1 the negative sign indicating that the corresponding 3) 123. ..1 weight 3^ is to be placed in the opposite scale to sTil. those indicated by the positive remainders. Thus, q fri 1 weights 39, 3^ 3% 1 must be placed in one scale 6)_Li...-L and 38, S', 36, 35 in the other scale. 3)_5...-l 3)_2...-l 1...-1 32. This follows from the fact that 33. Let the number be denoted by a.W^ + b.W-^ + c.lO'^^ + ...+p.l03 + q.W + r.lO + s; now 10^, 10^, 10^,... are all divisible by 8; hence the number is divisible by 8 if 3 . 102 + r . 10 + s is divisible by 8. 34. Since r = s - 1, the number nrr in the scale of s is equal to 10000 - 1, and the square of this is 100000000 + 1 - 20000 ; hence we have the result, siuce s-2 = q, and s-l=?'. 35. Let S denote the sum of the digits: then ^^ and ^'~^ are r-1 r-1 N ~N' both mtegers. [Art. 88.] Hence r- is also an iuteger. 36. Let 2ra denote the number of digits ; then the number may be represented by ar="'-i + Zw2"-2 + cr2»-3 + + cr^+ir+a. This expression may be written a (r'^"-! + 1) + 6r ()-2'»-3 + 1) + cr" (rS-'-e + 1) + . . ., and is therefore divisible by r + 1. VIII.] SURDS AND IMAGINAKY QUANTITIES. 29 37. It foUows from Art. 82 that — ^ is an integer; hence ^~^\ IS also an integer. Similarly ^ — ^ or — — ? is an integer. Hence ■ ^~ " is an integer. 38. The number mU be denoted by abcabc ; thus the number = a.l0« + &.10* + c.l03. +O.102+6.10 + C = a(103 + l)102 + 6(103 + l)10 + c{103 + l) = (103+1) (a. 102 + 6. 10 + c). Thus the number is divisible by 1001, that is by 7 x 11 x 13. This is a particular case of Example 40. 39. Let N be the number, S the sum of its digits, and r the radix ; then J/ - S = / (r - 1) , where I is an integer. But r - 1 is even ; hence N-S is even, and therefore N and S are either both even or both odd. 40. Denote ten by t, and let the number be p,t'^^+p^t"--' + +Pn-lt+Pn> on repeating the n digits p^, p^, p^, ...p^ the new number will be Pit^'^^+P2t^''-'+ ...+p„-.iV'+'-+pJ.-''+Pit^-^+p^P<--'+ ...+p^_jt+p^ = (ft*"-' +P^t'^'' + ... +P„-it+p„) «"+ (^i«"+i>2«''~''+ - +i'n-l« +Pn) = (i'l«''-'+i'2«'^'+ - H-Pn-l^+Pj («" + !)■ Thus the number is divisible by the original number and also by *•*+ 1. Also, since n is odd, i" + l is divisible by * + l, that is by eleven, and it can easily be seen that the quotient is 9090. ..9091; thus 100001 = 11 X 9091 ; 10000001 = 11 x 909091. EXAMPLES. VIII. a. Paqes 72, 7.3. ^ 1 l + s/2+s/3 _ l+V2 + V3 _ ;,/2 + 2+^6 ^' 1 + ^2-^8 (1 + ^2)2- (^3)2 2^2 - 4 • 2. x/2 ^ ,J%J2+s/3+^^ ^2+^3+^ 5 ^ V6+V3+>/15 ,^2 + ^/3-^5 (\/2+x/3f- 1^/5)2 2;^3 " 6 ' _ ^a+sjb-^a + b ija+s/b- ^Ja + b i^a+ijb+ja + b {s/a + ^b)''-{a + b) ijab __ aijb + bsja- Jab (a + 6) ~ 2a6 30 SUEDS AND IMAGINARY QUANTITIES. [CHAP. ^ 2 Ja + i ^ 2 Ja + 1 {^0^+ Ja + 1 +7 2a) ^ Ja-1 + Ja + l + iJ^ _ a-l + Ja^^+j2a{a-l) a-1 ^ ^^ . (^/ lO + V5 - >y3) (yiO + ^5 + V3) 5. The expressions ^^io + JS-^5){J10 + ^B+J5) (15 + 10^2) -3 _ (6 + 5^2) (V30 -4) ~(13 + 2;^30)-5 30-16 „ rru • 5+;V15 + ^/10+^6 6. The expressions ^2+^3+^5 (3^-11(3^-3^ + 1) 3-2.3* + 2.3*-l 13. The expression =^ —^ = ^ • 14. The expression = r- 3»+2t (3i-2^) (3^-3^.2^+3^.2^-3^.2^ + 3^.2^-2^) - 32-23 ' the denominator is unity, and the numerator gives the result. 15. The denominator is 3i'+22, hence as in the preceding example, the expression _ 2^ . 3^ (3^-3^ ■ 2^3^ . 2^- 3I . 2^ + 8^ . 2^ - 2*) 16. The expression = — 17, The expression = -23 3^ 1 3^-3^ + 3*-3^+3^-l 3^+3^- si^ + l 3-1 2^+2^ 2^ + 1 2f-2^ 2^-1 _ (2^ + 1) (2^ + 2^ + 2^+2^ + 2^ + 1) 2^-1 =^(25 + 2.2^^ + 2.2^ + 2.2^^1^+2.2^+2.2^ + 1). Vin.] SUEDS AND IMAGINARY QUANTITIES. 31 3* 3* 18. The expression 3-3^ 3T-1 32-1 Examples 19 to 24 are solved by the method of Art. 87; the results however may generally be vfritten down by inspection ; thus in Ex. 19 the quantities 20, 28, 35 under the radicals are the products of the numbers 4, 5, 7 taken two at a time; and the sum of these numbers is 16; .-. 16-2^20-2^28 + 2^35= (V5 + ;^7-V4)2; the two quantities ^5, ijl having the same sign, because of the term + 2^35. 21. 6 + V12-v/24-V8 = 6 + 2V3-2>y6-2V2 = (s/3 + l-^2)2 22. 5 - ^10 - ^15 + V6 = i (10 - 2^10 - 2^15 + 2^6) = ^ {^3+^/2 - v/5)'- 23. a + Zh + i + i^a-iJ^-2jSab = {^la- JZb + 2)\ 24. 21 + 3;V8-6v/3-6J7-V24-^56 + 2;^21 =21 + 2^18- 2^27- 2^63- 2^6- 2^14 + 2V21=(V9+v/2-;^7-,/3)2; the numbers 9, 2, 7, 3 are seen by inspection, and the signs before the radicals easily assigned by trial. 25. Proceeding as in Art. 89, we shaU find x^-y=. ^\00-WB = -2; and x^ + ixy = 10, whence a; = 1, 3/ = 3. 26. Here a;2- ?/ = 4/382 - 289 x 5= ;^1444- 1445 = -1; and a;' + 3a^ = 38; whence a;=2, y = 5. 27. Here x"- 2/= 4/980]T4900ir2= 4/1 = 1; and a? +3x2/ = 99, whence a; =3, y = %. 28. Here 38^14-100^2= -2^2(50-19^7); and a;2-2/= ^'2500-361x7= 4/"^^= -3; also a^+3x2/ = 50, whence a; =2, y = l; thus the cube root= -^2 {2-^l)=JU-2J2. 29. We have 64^3 + 41^5 = 3^3 ^18 + ^ y/|^; ' /oca 1681 5 7343" 7 here '^'-2' = \/ ^2*— 9" '^ 3= V 27" = 3 = 32 SURDS AND IMAGINARY QUANTITIES. [CHAP. 5 also x^ + 3xy = 18, ■whence x=2, j/=^; tlius the cube root=;^3 (2+ » /;t j = 2^3+x/5. 30. We have 135^/3 -87x/6 = 3^3 (45-29^/2). Here s2-y=^2025 - 841 x 2 = 4/343 = 7; and x^ + 3xy=i5, whence x=3, 2/ = 2; thus the cube root=;^3 (3-^2) = 3;^3-^6. Examples 31 to 34 may be solved by inspection ; thus 31. a + x+V2ax + ffi^=(ffi+x) + 2^|^a + |j = ^^|+ -y'a + l) ■ 32. 2a-V3a^-2a6-62 = l(4(t_2^(3a + 6)(a-6) 33. l+a2+^l + a2 + a4 = ^(2 + 2a2 + 2^r+a + a2)(l-a + a-)' = i (71+^+^+ Jl-a + ay. 34. l + (l-ar* = l+-^^ = -75— ^(2+2^1^=) 35. Here 0=2+^/3, 6=2-;^3; thusa + 6=4, a-6 = 2^3, a6=l. 7a2 + lla6-762=7(a+6)(a-6) + lla6=56v'3 + ll. 36. Here a;=5-2^6, y = 5 + 2^&; thus a; + 2/ = 10, X2/ = l. .-. 3a;2 - 5x2/ + 32/2 = 3 (x + j/)2 - llxy = 300 - 11 = 289. 37. Theexpression= ^^^^^^^ 3^3-5 10-^76 + 10^3 10-(5V3 + 1) ^ V3j-5_ J_ 9-5;^3~^3" 38. Dividing numerator and denominator by x/3, the expression under *^«-^-^ = i?^.= ^^^^^^^^f^^^^^ = S2 + B0^3 = (3V3 + 5). VIII.] SURDS AND IMAGINARY QUANTITIES. 33 39. The expression = (5 - v'3) - j^-m = (5 - \/3) - (2 - x/3). 40. The cube root of 26 + 15«/3 is 2 + ^/3. Henee the expression = (2+^3)2- (j^j^ = (2+^/3)2-{2-^3)'' = 4x2V3=8;^3. 41. Multiply each numerator and denominator by ^2 ; thus ,, . 20 2^5 + 6 the expression = , - ; 6-^/6 + 2./5 4+^y6-2^5 20 ■'-|i#=(5W5)-2. ~5-J5 3+V5 42. From the formula a' + b^ + c^-3abc = [a + b + c){a^ + b^ + c''-bc-ca-ah); we have a^ + 2-l + 3x4/2=x3+ (4/2)3+ (- 1)3 -3a;(4/2)(-l) = (a; + 4/2-l)(a;2+4/4 + l-x4/2 + a; + 4/2). 43. As in Art. 89 we have x' + 3xy = 9al^. Again {9ab^f - (6= + 2ia?)^ (6^ - Sa^) = 1728a« - 432o''62 + 860^64 _ (,6= (I2a2 - 62)3 . .-. !K2-2/ = 12a2-62; and thus 4x3 -3x(12a2-6'>)-9a62=0; or 4x(x2-9a=') + 362(x-3a)=0; whence x = 3o, y = b^-3aK 44. 4x-4=(^«+-iJ-4 = (>--iJ; .-. 2V^^=v/a-4-- , 1 -TV, .V, • ^"'^ «-l Thus the expression = ^ \ / i \ ~ ~2~ ' EXAMPLES. Vm. b. Pages 81, 82. 4. The produot = (x + w) {x + w^) =x^+(o>+ oi^ x + u^ = x'' - x + 1. = w V. 1 S + v'^ 3 + V^ 5. wehave _^ = ^-^-^ = _jj_ . fi The expression -^^^^±^>^- < V^ + ^'>/5)\ 6. Ihe expression - 3 ^2 - 2 ^5 ~ 18-20 H. A. K. 34 SURDS AND IMAGINARY QUANTITIES. [CHAPS. (3 + 2i) (2 + 5i) + (3 - 2i) (2-5i) _ 2(6 + 10f) ^ _ 8^ _ 7. The expression = 4 - ( - 25) " 29 29 {a+ix)^-{a-ix)'' iiax 8. The expression = ^^::^, ^^-r, ■ (x + if-ix-i)^ 2(3is2 + i3) 2i(3x2-l) 9. The expression = i ^^^ = ^^2 + 1 ^TiT ' 2(3ia2 + i') 30^^ + 1' Sa^-l 10. Theexpression= j^^— = —^^ = -^ • 11. (-V^)^»+3 = (_l)4>.+Sxj^— l)4n+3 ^ ^ = (-l)x(^/^l)' = (-l)x(-^/-l)=^/-l• 12. The square = (9 + iOi) + (9 - 40i) + 2 ^81 - IBOOi^ = 18 + 2.^/1681 = 100. Examples IB to 18 may be solved by the method of Art. 105, or by inspec- tion as follows. 13. -5 + 12 J^T= -5 + 2 J^M=-9 + 4:+2,J^^^i=U^+2y. 14. -11-60 V^=- 11-2 n/- 900 = _ 36 + 25 - 2 J-36x25 = (5 - J^M)^ 15. -il + sJ^S = -47+2j^i8 = (-48 + 1 + 2 V^^ = (1+n/^^'- 16. -8v'^=0-2 J^^=i-i-2 J -4.xi = {2- ,J^f. 17. a'-l + 2aJ^l = (a+J^\ 18. 4a5-2(a2_62)^rT=(a + 6)2_(a-5)2_2(a2-62) V^l = {{a + J)-(a-6)V^}'- iq We have 3 + 5i _ ( 3 + 5i) (2 + 3i) _ - 9 + 19 i 19. we have ^_^.- ^_^.^ - ^^ . 20. s/3-iV2 _ (^3-V2 )(2^ 3 + i;V2) _ 8-^6 2v/3-is/2 12-2i2 14 ^ , 1+i (l + i)(l + i) l + i2 + 2i . behave = — : = ^ — =^^-^5 — '-= = =i. 1 - 1 l-z^ 2 (l + i)^_l+i^ + 2i_ 2i _ 2i (3 + i) _ 6i + 2i^ _ 3i-l 22. 3_j 3_i 3_j 9_j2 10 23. Theexpression= (7;^);-/°-.;f = gi^ {a + ib){a~ib) a^ + b VTII., IX.J THE THEORY OF QUADRATIC EQUATIONS. 35 24. ■Wehavel + u2=-fc,; thae {l + t^y = {-a)*=a'>=a. 25. Wehave l-o} + J^ = (l + o) + oi^)-2u=0-2i,i=-2oi. Similarly l + (a-ur'=-2uy'. The product is 4u'=4. 26. Since 1-(o* = 1-(d and l-u'' = l -w^, the expression = (1 - u)^ (1 - u^f 27. 2 + 5ai + 2oj2 = 2 (1 + u + 0)2) + 3u = 3a>, and (3m)« = 729ai6 = 729. The solution of the second part is similar. 28. The factors are equal to 1 - u+.u^ and 1 - w^^ + u alternately, and the product of each pair is 2". Ex. 25. 29. x^ + y^ + z^-3xyz = {x+y+z) (x^+y^+z^-yz-ex-xy) = {x+y + z)(x + mj + by'z)(x + i^y + \ Hence (1) xyz = (o + 6) (o^ _ a6 + 6^) = ^3 + js. (2) x^+y''+z^=ai?+(y+zY-2yz = (a + hf+(a + l)f-2(a?-ab + V) = &ah. (3) x^ + y^+z^=3i? + (y+z)(y^+z>-yz) = ^ + {yJrz){(y + zf~^z} = (a + 6)3_(o + 6)j(a + 6)2_3{a2-a5 + 52)} = 3(a3 + 63). EXAMPLES. IX. a. Pages 88, 89, 90. 13. If the roots of Ax''+Bx + C=0 are real, then B^-iAG is positive. Now in (1) , 4a^-i (a" ~b^-c^)=ib^ + 4:C^, a positive quantity. Again in (2), U{a-b)^-4:{a-b + c){a-b-c) = 16(a-6)=-4(a-6)2 + 4c==12(a-6)2 + 4c2, a positive quantity. 14, Applying the test for equal roots to the equation x^-2mx + 8m-15 = 0, wehave m°=8m-15; that is (m - 5) (m - 3) = 0. 15, If the roots are equal (1 + 3m)' = 7 (3 + 2m) or Qm" - 8m - 20 ^ 0, that is (9m + 10)(m-2) = 0. 16. On reduction we have (m + l)a;^-6a;(m+l)=ox(m-l)-c(m -1), that is (m + l)a;2_{j(m + l) + a(m-l)}a; + c(m-l) = 0. The required condition is obtained by equating to zero the coefficient of x. 3—2 36 THE THEORY OF QUADRATIC EQUATIONS. [CHAP. 17. If the roots of Ax''+Bx+G=0 are rational, B^-iAC must be a perfect square. In(l), 4c2-4(c + o-6)(c-a + 6) = ic^ - 4(;2 + 4 (a - bf = 4 (a - 6)^ a perfect square. In (2), (3a2 + b^c^ - 4a6c« ( - 6a-' -ah + 26^) = c' {9a* + 2ia^i + lOa^b^ - 8a6' + b*) = c^ (Sa" + iab - 6^)2 = a perfect square. In Examples 18 to 20 we have be , b^-2ac a + 3=~-, aB=-; whence a^ + ^= — -j '^ a a «■ 1 1 _a= + |S2_62_2a£___c^_62j-2ac 19. a*^' + a7^«= a*/3^ (a^ + ,33) = a^ (« + ^) («' + ^' " «/3) ~a5V~oj a^ a' on /" ^y (a^-^)^ (g + /3)° (g - p)^ _ fg + j3)° Ua + ^)^ - 4a<3 } 20- V;8"gy ~ a>^^ ~ g2/32 g^^^ _j,2 /;,2_4ac\ c^ 6° (6' - iac) ~a^\ a^ J^^^~ aV 21. Form the quadratic equation whose roots are l±2i. This equation is a;'i-2x + 5 = 0. Therefore x'-ix + i is a quadratic expression which vanishes for each of the values 1 +2i, 1 - 2i Now x^ + x'^-x + 22 = x{x^-2x + S) + S{x^-2x + 5) + 7 = xxO + 3xO + 7 = l. 22. The equation whose roots are 3 ± i is ic^ - 6x + 10 = 0. Now ,i?-%x''-Sx + 15=x{x^-ex + 10) + 3(x'-Gx + W)-15=-15. 23. The equation whose roots are o (It J -3) is x^- 2aa; + 4a^=0. Now x^-ax^ + 2a'x + ia^=x{x^-2ax + ia') + a{x^-2ax + 4:a^)=0. 24. Here a+;8= -^, a/3=g. Sum of roots=:(g + ^)2 + (a-/3p=2(g2+p2) = 2(p2-2g). Product of roots = (g + /3)2 (a - /3)2 = j)2 (p2 _ ^g, ). 25. In the equation x^-{a + b)x + ab-h^=0, the condition for real roots is that (a + 6)2-4(a6-/i2) should be positive; that is, (o - 6)^ + 4^^ must be positive, which is clearly the case. 26. From the equation ax^ + bx + c = 0, we have ax^ + bx=-c, that is, ax + b= — =-c;tr'-; whence lax+b)~^=[-cx-'^)-'=— . IX.] THE THEORY OF QUADRATIC EQUATIONS. 37 In (1), (ax,+ 6)-H (a^, + 6)-=^^= (^i + ^2)^ - 2^,x, ^ 1 /6.^ _ 2o\ In (2), (ax, + b)-^ + {ax, + b)-^= _^i!+a'= _ 1 (^^ + ^^)(^^2 + ^^2_^^^j c' \ a) Xa' a J ' 27. Denote the roots by o and ma ; then a + na= — , and axna=-. a a Eliminating a, we have a'ii. + nf a' 28. Here aH^-5i=|^". and a- + ^-=^ = 5!z|££; hence sum of roots = ^4 , and product = ^ „ , ' . a-'c' ah' 29. Here o+|8= -(m + n), a^=- (m^ + n?). :. (a+^)2=(m + n)2; and (,a-^f={:m+nf-'i{jn^ + n'')^~{m-nf. Thus we have to form the equation whose roots are (m + b)^, -{jn-v)''; the sum of roots = 4mre, and product = - (m + m)^ (m - m)^. EXAMPLES. IX. b. Pages 93, 94 1. In the equation 2a'x'+2ancx+ (71^-2)0^=0, the condition for real roots is that aHh'^ - 26? (r^ - 2) c' should be positive, that is, 4 - m'-* should be positive. Therefore n must lie between - 2 and + 2. 2. Put -; — '- — -7:=y; t'henyx'^-{5y + l)x + dy==Q. Ifajisreal, {5y + 1)^ - 362/2 must be positive ; .■. (1 + lly) (1 - 3/) must be positive ; that is, y must lie between 1 and - iry. 3. Put |2-^^^=^; then (2/-l)a;2+(2/ + l)ii; + y-l = 0. Ifa;isreal, {y + 1)2 - 4 (2/ - 1)2 must be positive; .-. (y - 3) (1 - 3y) must be positive. 4. Put V+2a-7 "^= thenx2(y-l) + 2(y-17)x-7y + 71=0. II x is real, (2/ - 17)2 + (y - 1) (7^ - 71) must be positive ; .■. 8 (^2 _ liy ^ 45) must be positive ; .•. 8{y-5){y~ 9) must be positive. 38 THE THEORY OF QUADRATIC EQUATIONS. [CHAP. 5. Sum of roots = ^ + , = — ; rr=ir* Ja+^a-b ^a~sja-b a-(a-b) b Product of roots = ^ , x = =- . ^a + ija -b Ja- J a - 6 " Henoe tlie equation is ic^ — ;-a!+^=0. o 6. (1) o2(a2^-i-j3)+|82(/32a-i-a) op g ' ^ ' (2) From a;^ -pa; + 2=0, we have a- p=-; hence X (a;-i))-*=(ga;-i)-'=?j. Substituting o and p for x successively, 7. Denote the roots by pa and qa ; then n n pa + 2a=---; pax 50 = y. From the second equation a = -7^. . /-. Jpq Vl Substituting in the first equation ^tl . . /- + -=o. Dividing by ^ - we have the required result. — 2a -2m ^^' ^'^^^ '^^ + 2 (™ - 2/) a; + m^ - 4mre + 2m!/ = 0. Ifxiareal,(OT-j/)2-m2+ 4mn - 2ny must be positive ; .-. y^-{2m + 2n)y + imn must be positive ; that is, (y - 2m) {y - 2n) must be positive. 9. In the first equation we have a + 3= --2, a8=-- a '^ a' ... (a-^)==li^. IX.] THE THEORY OF QUADEATIC EQUATIONS. 39 Again, from the second equation we have {(„ + a)-(, + m==^^^^; thatis,(a-^)==ii?^; whence the result follows. 1°- ^"* ltt!-li =y' tten(^ + 42/)x''+3x(l-^)-(4+jp2/)=0. If a; is real, 9 (1 - y)^ + 4 (p + iy) (4 + py) must be positive ; .-. (9 + 16^)) J/" + 2 (2^)^ 23) ^ + (9 + 16^)) must be positive ; .-. {2p'+ 23)2 _ (9 + 16p)2 must be negative or zero, and 9 + 16p must be positive. Thus 4 (j)2 + 8p + 16) {p^-Sp + l) must be negative or zero ; that is, 4 (j) + 4)' (2) - 1) (p - 7) must be negative or zero. "■ ^"' 2a:43x+6 =^' tli^n 22/:.=i + (32/-l)-^ + 62/-2=0. If X is real, (3^ - 1)^ -8y{&y-2) must be positive ; . . (1 + 13y) (1 - Sy) must be positive. Hence y must lie between ^ and - =-j , and its greatest value is 5 . 12. Put I 7 =y; then siP-2yx + 'by + cy-ic=0. If a; is real, y'-by -cy + bc must be positive ; .•. (y-b){y-c) must be positive. 13. In order that the roots of ax^ + 2bx + c=Q may be possible and different we must have b^-ac positive. The second equation may be written (a^-ac + 2b^)x'' + 2b{a + c)x + c^~ac + 2b^=0; and the condition for roots possible and difierent is that b' {a + c)2 - (a2 -ac + 26^) (c^ _ac + 262) should be positive. This expression reduces to (ac - 6') {462 + (a - c)^} , so that its sign is contrary to that of 6^ - ac. Hence the required result follows at once. 14. Denote the given expression by y ; multiply up and re-arrange, then (ad - bey) x'^-(ac + bd) {i.-y)x+ {be- ady) = 0. If X is real, we must have (ac + 6d)2 (1 - 3/)2 - 4 {ad - bey) (be - ady) positive ; that is, { (ac + bd)^- 4a6c(i} (y^ + l)-2y {(ae + bd)" - 2 (aH' + bV) }, or (ae - bd)^ y^ - 2y {(ac - bdf -2 (ad- bcf\ + (ae - 6d)2 must be positive for aU values of y. 40 THE THEORY OF QUADRATIC EQUATIONS. [CHAP. Thla will be the case provided (ac-6d)*>{(ac-M)2-2(a(J-Jc)2}2, that is, (ac - UY> (ac - bdy - 4 (ae - id)" {ad -bo)^+i{ad- 6c)*, that is, {ac-id)^>(ad-bc)^; that is, (ac -bd-ad + be) (ac-bd + ad- be) is a positive quantity; .-. (a + b){c- d) (a -b)(c + d), or (a« - If) (c" - d') must be positive. Hence a" ~ V and c^ - d^ must have the same sign. EXAMPLES. IX. ff. Page 95. Questions 1 and 2 may be solved by application of the formula of Art. 127. 1. Here m- 1 + 3 = 0, whence m=- 2. Or thus: the given equation may be written 2x{y + l) + y^+my-3=0; hence y + 1 must be a factor of y'+my -3; that ia,y=-i must satisfy the equation y^+my-3=0. 25 m^ 2. Here the condition gives-12- — + — =0, whence m'=i9. 3. The condition that the roots of Ax'^-{'B-C)xy-Ay'^=(S should be real is that (B - Cf-^iA? should be a positive quantity: this con- dition is clearly satisfied. 4. Since the equations are satisfied by a common root, we must have (a;2+^a; + 2)-(a;2+ya; + 5') = (1). Also by eliminating the absolute term, we obtain g'(a;2+px + 2)-}(a;2+p'a; + 3') = o (2;. From (1) we get x=^^' , and from (2) z=- ^^'~^'^ . 5. When the condition is fulfilled, the equations h? +mxy + ny^=0 and l'x^ + m'xy + n'y^=Q must be satisfied by a common value of the ratio x : y. From these equations we have by cross multiplication _ ^y ^ mn' - m'n nV - n'l Im' - I'm ' whence (nl' - n'l)' = {mn' - m'n) {Im' - I'm). 6. Applying the condition of Art. 127, we have 6- 4aP- 12- 2a2-p2=0. IX.] THE THEORY OF QUADRATIC EQUATIONS. 41 7. If 2/ -ma; is a factor of ax^+ihxy + hy^, then tins last expression vanishes when 2/ = ma;; that is, a + 2hm+bm,^=0. Similarly iimy + xisa, factor of a'x^ + 2h'xy + h'y^, we must have a'm'-2h'm + b' = 0. From these equations, we have by cross multipUcation mi' _ m _ 1 2{b'h+ah') ~ aa" - W ~ -2(bh' + a'h) ' whence {aa' - bby= - 4 {ah' + b'h) (a'h + bK). 8. Here x^-x(Zy + 2)+2y^-Zy~Z5 = 0; whence solving as a quadratic in x, 2ar=32/ + 2±v'(32/ + 2f-4(2j/2-3j/-35) = 32/+2±(2/ + I2). Giving to y any real value, we find two real values for x: or giving to x any real value we find two real values for y. 9. Solving the equation 9a^ + 2a (y- 46)+;/^- 20y+ 244=0 as a quad- ratic in a, we have 9a;=- (2/ -46)±7(j/ -46)2-9 (3/2-20!/ + 244> = -(y-46)± V-8(j/2-llt/ + 10) = -(2/-46)±^/-8(2/-l)(y-10). Now the quantity under the radical is only positive when y Ues between 1 and 10; and unless y Ues between these limits the value of x wUl be imaginary. Again j/2+2?/ (a;-10) + 9a;2-92ar+244=0; whence y=-{x-10)± J{x- lO)^ - (9x2 - g2x + 244) = -{x-10)J= J -8 {x-6) {x-3}. Thus in order that y may be real x must He between 6 and 8. 10. Wehave x^{ay + a')+x{by + b')+cy + c'=0; solving this equation as a quadratic in x, 2 {ay + a') x= - (by + b')±J{by + b'f-4:{ay + a') {cy + &). Now in order that x may be a rational function of y the expression under the radical, namely (62-4oc)!/2 + 2(66'-2ac'-2a'c)y + 6'2-4a'c', must be the square of a linear function of y; hence {bb' - 2ac' - 2a'c)^ ={b^- 4ac) (6'2 - 4a'c') . Simplifying we have a'c'^ + a'^a^ - ac'bV - a'cbV + 2aa'cc'=4,aa'cc' -acb'"- a'c'b^ ; .-. a^c'^ + o'%2 - 2aa'cc' = ac'bb' + a' ebb' - acb'^ - a'c'b' ; .-. {ac> - a'cf={aV - a'b) {be' - b'e). EXAMPLES. X. a. Pages 102, 103. 1. (a-i - 4) (ic-^ + 2) = ; whence - = 4 or - 2. 42 MISCELLANEOUS EQUATIONS. [CHAP. 2. (a;-2-9)(a;-2-l)=0; whence i = 9 or 1. 3. (2s4 - 1) (a;4 - 2) = ; whence \/a; = - or 2. 4. (3a;4 - 2) (2a;4 _ 1) = ; whence ijx=-^ or 1 11 i_ i 5. (a:»-3)(a;'*-2)=0. 6. (a;2»-l) (a;2»-2) = 0. 7. Putting y= x/o> we have 7y + - = -^-; whence y=ar °^ ^• 2 2/^6' "-"""' !>-2 " 3' 8. Putting y= \/ YZT ' '^^ ^^™ 2/ + - = -^! whence 2/=^ or 9. (Ss^-l) (2a;i + 5) = 0; wh6nce;^x=5 or -5. 25 The value x=-^ satisfied a modified form of the given equation. 11. (3>^-9)(3»'-l)=0; whence 3=^=9 or 1. 12. (t.5'-l) (5»-5)=0; whence 5*=|=5-i, and 5»^=5. 5 13. 22=^8 -2. 2"=+* + 1 = 0; that is (2"^i- 1)2=0; whence a; + 4=0. 14. 8.22»^-65.2=: + 8 = 0; that is (8 . 2»'-l) (2»'-8) = 0; whence 2^ = 3 = 2-«, and 2== = 2'. 15. (v/2^- 1)2 = 0; whence ^2^=1, and 2"^ = 1. . 3 v 59 1 16. Putting y = J2x, we have --5 = 10; whence y = ^ot - 30. 17. (x-7)(a; + 5)(a;-3)(^ + l) = 1680; that is, (!<;2-2a;-35)(a!2-2a!-3) = 1680; this is a quadratic in a? - 2a; and gives (os^ - 2a! - 63) (a;^ - 2a! - 25) = 0. 18. (a;+9)(a:-7)(a!-3)(a! + 5) = 385; that is, (a:2 + 2a;-63)(x2 + 2a!-15) = 385; this is a quadratic in x^ + 2x and gives (x^ + 2a;,- 70) (a;^ + 2a! - 8) = 0, X.] MISCELLANEOUS EQUATIONS. 43 19. K (2a! -3) (2a! + !)(»- 2) = 63; that is (2a;2-3a:)(2a:2-3x-2)=:63; this is a quadratic in 2x'' - 3x and gives (2a:'' - 3a; - 9) (2x^ - 3a! + 7) = 0. 20. (2x-7)(a! + 3)(x-3)(2a! + 5) = 9l4 that is, (2a:2-a!-21)(2a;2-x-15) = 91; this is a quadratic in 2a:'' - x and gives (2x' - x - 8) {2x' - x - 28) = 0. 2L Put2/2=x'' + 6x; then ^2 + 22^-24=0; thuS2/=4 or -6; and x2 + 6x = 16 or 36. N.B. In this and the following examples, the solution obtained by taking the negative value of y satisfies a modified form of the given equation. 22. Put 2/2 =3x2 -4x- 6; then j/^ + j/- 12=0; thus2/ = 3or -4; and therefore 3x''-4x-6 = 9 or 16. 23. Put y^ = Sx^-lGx + 21; then 1/2 + 3^-28=0; thus 3/=4 or -7; and 3i2-16x + 21 = 16 or 49. 24. Put ^2=3x2-7x + 2; then y'-3i/-10=0; thus y = 10 or -1; and 3x2 _ 73.^ 2 = 100 qj. i_ 25. Put ^2=2a:2-5a; + 3; then y^-6y + 5=0; thus y=l or 5; and 2ii?-5x + 3=l or 25. 26. Put 2/2=3x2 -Sx + l; then 2y'^-y-6G=0; thus 2/ = 6 or --g-; and 3x2-8x + l = 36 or — . 27. Dividing by ^x-3, we have ^/x -3=0, and .^4x + 5 - ^x = Jx + S; then see Art. 131. 28. Dividing by s/2x - 1, we have V2x-1=0, and ^x - 4 + 3= ,Jx + Ti. 29. Dividing by ^/x - 1 ; we have »yx-l=0, and v/2x + 7 + v'3(x-6) = Jlx + T. 30. Dividing by ija + 3x, we have Ja + Sx=0, and Ja-x-Ja-2x=sj2a-ix. Examples 31 to 34 may be solved as in Art. 132. 31. Use the identity (2x2 + 5x - 2) - (2x2 + 5x - 9) = 7. 32. Use the identity (3x2-2x+9)-(3x2-2x-4) = 13. 33. Use the identity (2x2 _ 7^ + 1) - (2x2 - ga: + 4) = 2x - 3. 34. Use the identity (3x2- 7x-4) _ (2x2- 7x + 21)=x2- 25. Examples 35 — 37 are reciprocal equations and may be solved by the method of Art. 133. Example 38 may be solved by Art. 134. 44 MISCELLANEOUS EQUATIONS. [CHAP. a: I 39. We have componendo et dividendo — p- — v H-3 ; thus xi»= 2a! + 3. 45. Dividebya"^; then a"^ (0^ + 1) = (a'": + 1) a; VoMiia.a^'^-a,'' .a'-w' + a=0; that is (a . a"' - 1) (a"^ - a) = ; whence a'=- = a-'; and a^=a. a 46, Clearing of fractions, 8(a;-5)2^=(3x-7)^; taking the cube root of each side, 2tjx-h= JSx - 7. 47, The solution is similar to that of Ex. 46. 48, Dividing each term by (a^ - x^)^, or (a + ar)3 . (a - x) 3, we get fa+x\l ^Ja-x\i \a-xj \a + xj or 2,+f=5, where j/ = (^)\ X.] MISCELLANEOUS EQUATIONS. 45 49. We have identically (x^ + ax -1) - {x^ + bx -l) = {a-b) x; and by the question , ^x'^ + ax-l-Ajx^ + bx-l= ija -^b: hence by division, Jx^ + ai - 1 + Jx'^ + 6x - 1 = {Ja +,Jb)x. By addition, 2 Jx'' + ax-l = {Ja + Jb) x + {Ja - ^b). Squaring, 4 (x^ + ax - 1) = {Ja + ,Jbf x^ + 2{a-b)x + {^a- ^Ib)^; ■■■ {(^/a + ^/6)»-4}x2_2(a + 6)x + {(^a-;^6)2+4}=0. Now by inspection, the original equation is satisfied by a;=l; hence by the theory of quadratic equations the other root is j^. '^^jjr^ — ^ ■ 50. The simplified form of the left side is 2x^+2 (x^ - 1) ; thus 2x2 + 2(x2-l) = 98. 51. This equation may be written x*-2x'+x''-x''+x=380; that isa?{x- 1)" - x (x - 1) =380 ; which is a quadratic in x (x - 1). 52. This equation may be written 27x* + l + 21x + 7=0, that is (27xS + l) + 7(3x + l) = 0; dividing by 3x + l, we have 3x + l=0, and 9x2-3x + l + 7 = 0. EXAMPLES. X. b. Pages 106, 107. 20 , o 40 „ 1. «= — : hence 3x =7. ■' X ' X 2. j/=5x-3; hence (5x-3)»-6x'=25. 3. 4x = 3^ + l; hence 3?/ (3?/ + 1) + 13^2 -25. 4. By division lfi + xy+y''=i^•, combine this with !iP-xy+y^ = lQ. Examples 5, 6, 7 are solved by the method of Ex. 1, Art. 136. Examples 8 to 12 : transpose if necessary; the equations will be found to be homogeneous, and may be solved by putting y=mx. Examples 13 to 15 may be solved by the method of Ex. 2, Art. 136. 4 4 4 16. From (1), y=:j-— ; hence =-— + - = 25. 17. Prom (2), x+y=3 ; from (1), 2(xS+yS)=9xy ; by division 2(x2-x!/ + j/2) = 3x2/, or 2x'-Sxy + 2y'-=0; whence (2x-y) {x-2y)=0. 18. Put ^=u, %=v; then u + v = 5, and - + -=-;; whence we have " 2 5 u D 6 uv = 6. 19. Put u=x^, t)=j/»; then the equations become m'+«5=1072; tt + D = 16. 24, 46 MISCELLANEOUS EQUATIONS. [CHAP. 20. Put u=xi, v=yi; then the eijuations become «^u+«i)''=20, and u^+v^=65. Multiply the first of these by 3 and add to the second ; thus {u + v)^=: 125 ; whence «+?; = 5. 21. Put u=x^, v=yi; then the equations become u+v = 5, 6(-+-|=5: whence we find Mu = 6. \u vj 22. Square the first equation; thus 2x + 2 ijx''-y''=16; substituting from the second equation 2a; + 6 = 16. 23. Square the second equation; thus 2a; -2 »Jx^-l=y = 2- .Jx^-l from the first equation ; hence tjx" - 1 = 2 (a; - 1). ;. The first equation is a quadratic in * /-, and gives \/ y = ^ or 5 ; whence x=9y, or x=^ . 25. The first equation is a quadratic in —. ^^ , and gives ~ — ^=4 ^x-^y' ° ^x~^y 1 ti, * • \/^ 5 5 , a; 25 or -r ; that is ^ = ;; or — - ; whence - = -— . 4 v/2/ 3 -3 2/9 26. Multiply the second equation by 4 and add to the first ; thus {a;2 + 4^2/ + 43/2) - 15 (a: + 2)/) + 56 = 0. This is a quadratic in x + 2y, and gives x + 2y = 7 or 8. Combine each of these separately with xy=8. 27. The first equation is a quadratic in xy, and gives xy = 25 or 16. From the second equation (x - y) (4a; -y) = 0. 28. From the first equation, (2a; -Sj/)^- (2a; -5)/) -6=0. This is a quadratic m2x-5y, and gives 2a; - 5y = 3 or - 2. Combine with the second equation. 29. From (1), (3a;-2!/)2 + ll(3a;-2!/)-12=0; whence 3x-2y = l or - 12. Combine with the second equation. 30. Divide (2) by (1); thus {x''+y^){x + y) = iOxy: divide this last equa- tion by (1); thus , _ % = 7^ = 2' *'^°'' '^' Sx^-'i-0xy + 3y^=0; whence (Sx-y){x-Sy) = 0. Thus x=32/ or |. Substitute in the first equation. o 31. By division ^^J^/ ^l^i' *^^* "' 6:^'=-13^2' + %= = 0; whence (2a;-j/)(3a;-5i/) = 0. Substitute !i;=| , and a;=-J^ successively in the second equation. X.] MISCELLANEOUS EQUATIONS. 47 32, From (1), x^-xy + y^ x^+xy+y' 43a; 2x(x'+2y^) 43a; x + y x-y 8 3?-y^ 8 ' whence a;=0, or 9a:?=25i/i'. Substitute a;=±-^ in the second equation; a;=0 gives no solution. 33 and 34 are solved by the method of Ex. 4, Art. 136. 33, Here^!^gf^i^) = ^=3; thus m3-6m2+ 11m- G = 0; that is, (m-l)(m-2)(m-3) = 0. 34, Here ~ ™_\^ = -^=-21; thus mS-8m= + 21m- 18=0j that is, (m-2)(m-3)(m-3) = 0. 35 and 36 are solved by the method of Ex. 5, Act. 136. 35, From (1), a;*-9a;2/3_4x8j/i = _108j/2=-2/2(2a;!! + 9an/ + 2/2), by (2). Thus x^-ix^y^ + y^^O; that is (x^-y^f=0; whence x^-y'^=0; that is x=J=y. 36, From (1), (6a;<+a;V -2a;!/S)- (4 x6a;2) + (4)2=0; substituting from (2), namely i — x'^ + xy-y^, we have 6x* + a;y - 2a;j/5 - 6a;2 (a;2 + a;j/ - 2/») + (a;2 + x)/ - j>«)2 = 0, whence !(;*-4a;'y + 6a;'!/2-4aa/5 + 2/*=0; that is {x-yY=0, and a;=y. 37, From (l),x^-y^=hy-ax; dividing by (2), ^-^ = f^""^ ; whence ^ a; + 2/ by + ax - = — ; that is, -77=* -?-=&, say. Substitute in either of the given y ax s/^ Ja ' •' ^ equations, 38, Square (1) and subtract (2); thus 2a6a;t/=4aV-2iV; that is, 2a'x^-al}xy-iy=0, or (2ax+by){ax-hy)=0. Thus x=-^, 01 x = ^. Combine each of these with the first of the given equations. 39, On equating the first expression to and simplifying, we obtain b^x + a?y = a'h + dbK Similarly from the second expression we find xy -bx - ay + a? - ab + h^=0. Substituting for y from the first of these equations in the second, we obtain 6W-2a62a;_a3(a-2J)=0, whence (bx-a^){bx + a(a-2b)) = 0. h^x^ 106a; + Saw 40, Divide (1) by (2), and we get ^ = ^^^^^3^^ ; whence, by putting , 6a; ti • a lOm+3 m for — , we obtam m»=i-= — =— ; ay 10 + 3)71 3m4 + 10m3=10m + 3, 3(m*-l) + 10jrt(m2-l) = 0; whence m?-l = 0, or 3m2 + 10m + 3=0. 48 MISCELLANEOUS EQUATIONS. [CHAP. 41. From(l), we liave 2ax^ + {ia^-l)!cy-iay^=Q; Substitate these values in the second equation. EXAMPLES. X. c. Pages 109, 110. 1. From <1) and (2) by cross multiplication, x_y_z_ 2. From (1) and (2) by cross multiplication, X _ y _z 3. From (2) and (3), {x-y)''-z^=12; putting u=x-y, we have m2_22=i2. Also from (1) u-z = 2. Whence m=4, z = 2; thus x-y=^. Combine with xy = 5. 4. From (2) and (3), (x-zf-hi^= -11:, putting u=x-z, this gives iy^-u^=ll; also from (1), 2i/ + M=ll; whence 2y = Q, and «=5. Thus x-z = 5. Combine with a;2 = 24. 5. From (1) and (2), (x+yf-Sz (x+y)-z^=Z; putting u=x+y, this gives v?-^uz-z'^=i. Also from (3), u-z = 5. These equations give 11 4 z=2 or , and m=7 or ^ . Combine these results with the first equation a;2 + 2/2-«« = 21. 6. By addition of all three equations, x^+y^ + z^ + 2xy + 2xz + 2yz = %%; that is {x+y + zf = %&, and x + ?/+«=±6. Divide each of the given equa- tions by this last result. 7. The given equations may be written x(a; + 2j/ + 3z) = 50, y (x+2y + Sz) = 10, z(x + 2y + Zz] = 10. „, X y z X y z , ^""^ 5o=K=io°'^ 5=1=1="' ^^y- Or, multiply the second equation by 2, the third equation by 3, and add to the first ; thus (x + 2y + ^zf = 100. 8. Put u=y-z, v=z + x, w=x-y; then Mt7 = 22, ?;«; = 33, «)M=6; thus „2^2j(,2_22x33x6; whence uvw= ±66, andM= ±2, d=±11, «>= ±3. 9. By multiplication, !i!VsW=128 = 2'; thus xyzu=2. Dividing each of the given equations by this last result, we have xyz = b, xyu=i, xzu=-, yzu=-. Now divide the equation xyzu—2 by each of these four equations. X.] MISCELLANEOUS EQUATIONS. 4.9 10. Divide (1) by (2), thus i^ = ? . Multiply (1) by (2) and divide by (3), thus -=9. Substituting in (1), z" =3x9^^=3"; whence 2=8. 11. These equations may be written (x + l){y + l) = 24t, {x + l){z + l)=i2, (i/ + l){z + l) = 2S. Multiplying these together and taking the square root, we have (x + l)(^ + l)(z + l)=±168. Divide this result by each of the three equations above. 12. These equations may be written {2x + l){y-2) = 15, (y-2){3z + l) = 50, (2x + l)(3z + l) = 30. Whence (2a + l){y- 2) {3z + 1) = ± 150. Divide this result by each of the three equations above. 13. From (1) and (2), xz + yz + x + y = 15z; that is (x+y) {z + l] = 15z. Combining with (3), (12-a) {z + l) = 15«, whence z = 2 or -6. Substitute these values of z successively in the equations x + y = 12-z and xz + y = lz. 14. Subtract (2) from the square of (3), thus yz+zx+xy = (i (a). Subtract (1) from the product of (2) and (3), thus y^z + yz^ + zH+zx^ + xhj + xf'^O (^). Combining (1) and (a), we have {^+y + z) (yz+zx+xy) = 0. Subtracting (/3) from this last result, we have Zxyz=0. Hence one of the quantities x, y, or z must be zero. Let x=0; substitut- ing in (a), we have ys=0; thus a second of the quantities must be zero. Hence from (3) the remaining quantity must be equal to a, 15. From the first two equations, we have »" + 3/2 + z2 + 2 (y z + OT + a!y) = 3a2 ; that is, x+y + z==i=atJ3. From (3), 3x-y+z = a^3. I. From x+y+z=asJ3 and 3x-y+z = a^3, wehave 3/ = ":, z = aJ3-2x. Substituting in the first equation, we find 3x^-2^3. ax + a^=0, or (x^3-af=0; a that is, x=-y=. H. A. K. 4 50 MISCELLANEOUS EQUATIONS. [CHAP. II. From x+y + z= -a^Z a,nA. ix-y+z=aiji, we have y=x-aiJZ, z= -2x. Substituting in the first equation, 3x= - J^ax + a?=0, whence - = 's/^'^V-^ ^ a o 16. From the first and second equations, ^ + y^+z^-2yz- 2xz + 2xy = 9a^, that is, x + 2/-z=±3o. I. From x + y-z = 3a ani 3x + y-2z=3a, we have y = 3a + x, z = 2x. Substituting in the first equation, we have x'' + ax-2a'=0, whence a: = a or -2a. II. From x + y -z= -3a, ani Sx + y-2z = 3a, we have y = x-9a, z = 2x-Ga. Substituting in the first equation, we have x2 - lax + 16a2 = 0, whence - = ''^'J-'^^ . EXAMPLES. X. d. Pages 113, 114. 1. Divide by 3, then x + 2y + ~ = 3i + -^; thus -^^ — =mteger; multiply by 2 ; thus j^ + -^— = integer ; that is ^-5— =p ; hence y = 3p + 2 and ii;=29-8p. tus 2x hence as =2jj + l, y=24:-5p. 3, Divide by 7, then x + 5y + -^ = 21+-; thus -^= — = integer, and V-1 therefore ^-=— = integer =p say ; thus y=7p + l, and a; = 20 - 12p. 2x 7 2x — 7 4, Divide by 11, then x+y + j^=31 + YT; thus ^pj— = integer ; multiply ic — 9 ic — 9 by 6, then x- 3 + ^pj—= integer; that is ^pp=^; hence x=9 + llp &nd.y = 27 -13p. 5, Divide by 23, then ''+'!/ + ^=^^ + oS'> ^^^^ ^~ = integer; and v — 9 therefore ^g=-= integer =i) say; thus^=9 + 23p, a; = 30-25^. 2. Divide by 2, thus 2a; + 2/ + ;; = 26 + ^; therefore —^ - = integer =p say ; X.J MISCELLANEOUS EQUATIONS. 51 6. Divide by 41, then x+y + ^=53 + ~; tlius ?^^ = integer, and w — 3 therefore ^j-- = integer =2) say; thnsy = S + ilp, x=50-i7p. 7. Divide by 5, then * - 2/ - "^ = ? ! t^ua ^^ = integer ; multiply by 3, then 2/ + 1 + L^= integer; thus ^^=p, or y = 5p-i, x = lp-5. 8. Divide by 6, then a; - 22/ - 1 = , , thus ^-^ - integer =p ; hence 2/ = 6p - 1 and a = 13^ - 2. 9. Divide by 8, then x-2y--^ = i+-; thus °^ = integer; multiply by 5, then Sjz + ^-g- = integer; thus ^i-5- =p, or 2/ = 8»-5, a!=21«-9. o O 10. We have at once — = -^=p say ; thus x = lip, y = 13^. ix 7 4s! + 7 11. Divide by 19, then y-^-Tq=jqj tli"S -= integer; multiply by 5, then a; + lH — =-5— =uiteger; thus =j); hence x=19p-lQ and ?/ = 23p - 19. 12. Divide by 30, then 22^ + ^-^ = 9+|g; tlius i^^=^ = integer; multiply by 7, then 4y-5- „ =uiteger; that is i-—-—=p, or ^=80p-25, !B = 77p-74. 13. Let X be the number of horses, y the number of cows; then 37a; + 231/ = 752. 14a; 16 14a!— 16 Divide by 23, then x + y + -^ = 32 + ^^ ; thus — =j — = integer, and there- 7a; — 8 a: — 11 fore = integer. Multiply by 10, then 3a;- 3h — ;r-- = integer ; thus a; -11 =p, and the general solution is x = 2ip + U., y = 15-S7p. 14. Let X denote the number of shillings, y the number of sixpences; then 2a! +2/ =200; here x may have aU values from to 100, and therefore the number of ways is 101, 15. A multiple of 8 may be denoted by 8x, and a multiple of Shj 5y; thus the two numbers may be denoted by 8x and 5^; then 8x + 5y = 81, The general solution isa; = 5p + 2, y = lS-8p. 4—2 52 MISCELLANEOUS EQUATIONS. [CHAP. 16. Let X be the number of guineas paid, y the number of half-crowns received; then reducing to sixpenny pieces, we have 4:2x-5y = 21; the general solution is a;=5p + 3, y = 21p + 21. 17. Let X and y represent the quotients of the number by 39 and 56; then the number = 39a; + 16; and the number also = 56?/ + 27 ; hence 39a; + 16 = 56y + 27, or 39x - SGy = 11. Divide by 39, then « - 3/ - ^ = 39 : *li"s ^^ = integer ; multiply by 2, then 2/ + ??^^= integer; that is —^^= integer; multiply by 8, then 4_j, + ?^ = integer; thus ^^=P, or y = S9p + 20, a;=56p + 29. 18. Let X be the number of florins paid, y the number of half-crowns V 1 V -hi received; then ix-5y=5S; thus x-y-j=ld+-T; and therefore 2__ = an integer =p; whence the general solution is y = ip-l, x=5p + 12. 19. Let X denote the quotient of the part divided by 5, and y that of the pajrt divided by 8; then the two parts may be represented by 5a; +2 and 8y + S. Thus (5a; + 2) + (81/ + 3) = 136, that is 5a; -I- 8?/ =131. The general solution is x=2S-8p, and y=5p + 2. 20. Let X, y, z denote the number of rams, pigs, and oxen respectively; then we have x+y + z = M), a-ai 4a; + 2j/ H- 17« = 301. Whence 2x -^ 15« = 221. The general solution is a;=15^H-13, = 13-2p; whence y=li-\Zp. 21. Let X, y, z denote the number of sovereigns, half-crowns, and shillings respectively; then we have x+y + z = 27, and 40x-l-5j/-t-23=201; whence 38x+3y = U7. The general solution is a; =3^), y = i9-38p; whence z=35p-22. EXAMPLES. XI. a. Pages 122, 123, 124. 5. Wehave 4» (B-l)(m-2) = 5 (re-1) (m-2 (m-3); .•. 4»=5(b-3); .'. »=15. 6. The number=|8 vfithout restriction; if t and e occupy specified places, we can arrange the remaining letters in 16 ways. 7. The number =^0^=15; if each such selection is arranged in all possible ways to form a number, we get 15 x 14 = 360. 2n(2m-l)(2»-2) _44„ >i(n-l) . ^- ^^^ 1.2.3 ~ 3 1.2 ' whence 2m -1 = 11, or n=:6. XI.] PERMUTATIONS AND COMBINATIONS. 53 10. We can now only change the order of 6 bells, therefore the no. of ohanges=|6 = 720. 11. The number of ways =''^C?4= 10626. When the particular man is included we have to select 3 men out of the remaining 23 ; this can be done . 23.22.21 ,„„, m -^ g g , or 1771 ways. 12. Suppose the letters a, u fastened together ; then they count as one letter and we have six things to arrange. This can be done in 720 ways; but since a, u admit of two arrangements among themselves we must multiply this result by 2. 13. The number =^^0^ x i»C3=6375600. 14. (1) There are 3 ways of choosing the capital, and then |5 ways of arranging the other letters; therefore 3x |5, or 360 is the no. of~arrange- ments. (2) The no. of ways of placing the capitals at the beginning and end is 3 X 2 ; and the remaining letters can then be arranged in 14 ways ; .•. no. of arrangements=6 x24 = 144. 15. wC4e=«°C4=230300. 16. We have 12 + 8=7i by Art. 145; .-. m=20, and ^^G^^, "-^G^ may be easily found. 17. Here we have 3 places in which two letters are to be placed; this gives rise to 3 x 2 or 6 ways. Then the four consonants can be arranged in 1 4 ways ; .•. required no. of ways = 6 x 24 = 144. 18. (1) 4x8C, = 4xi^ = 224. (2) We must have 1 officer and 5 privates, or 2 officers and 4 privates, and 3 officers and 3 privates, or 4 officers and 2 privates; .■. the required no. of ways is 4 X 8C5 + ■'C2 X 8C4 + 0(73 X ^Ca{x^-a')x^ + ^CsX^} = 2 {5x5 _ I0a2x3 + Sa*x + lOx^ - IQaV + x^ = 2(16z6-20oV + 5o*x). 23. Thevalue = 2{G(V2)' + 20(V2)3 + 6v'2} = 12x4^2 + 80V2 + 12v'2 = 140^2. 60 BINOMIAL THEOREM. [CHAP. 24. Thevalue=2{2i!+15.2*(l-x) + 15.2='(l-x)i'+(l-a;'} = 2 {64 + 240 - 240x + 60 - 120x + 60a;2 + 1 - 3a; + 3a;2 - a;3} = 2{365-363a; + 63x2-a;2}. 25. There are 11 terms in the series ; .-. the middle term is the 6"'=i''C75=252. / x'^V 14.13.12.11.10.9.8 x" 26. The 8'" term = ^*Oj--^j = i .2 . 3 .4. 5 .6. 7 ' 128 429 16 "" 27. The expression = a'' .: in the expansion of ("1+-^) we have to find the coefficient of ar""; this is equal to i5C4(3a)*=110565a*. 28. The expression = al>x^^ (l 3 j , and the required coefficient / 1 \^^ 29. Since the expressions x^" I 1 - -^ 1 , we require the coefficients of x-28 and x-" in the expansion of {l-x~^)^\ these are ^^G^ and -^'Cu respectively. Thus the coefficients required are 1365 and -1365. / o'N* isq 30. The5"'term=''0,(3a)ii^-g j = -g^ a"'- (a^\ 5 21 /3 ,\V, 2 \9 5»x^^ /, 2 \9 31. The expression =(^ga;2j [^- W') = ^ "" [}- 9^') ' .-. the term required =^Ce\^-^j . ^ = .^ g 3 ■ 33^3 = jg "Vi) = i.2.3." 12 18.17.16. 15.14.1 3 4.5.6 = 18564. 3. Let the {p + l)"' term be the one required; then "Cpa:"-*. (-J , or "Cpa:""^" is the term containing x'. Therefore m - 22) = r, or p = - 33. n-r T"' I" .-. the coefficient=''Ci,= '" \i(n-r)\i(n + r) - XIII.] BINOMIAL THEOEEM. 61 [ IN'" / IN'" /IN" 34. Km-— A = ai'" ( 1 — ^ J , and we require the ooefflcieut of I — 1 (1 N^" 1 g I . Hence the required term = (-!)» . ' — [2b" 35. Let the (r + 1)'" term contain x". Then 2»c, (a;2)2''-r _ i-^, or *'C^*'-3'- contains x^; therefore 4ra-3?-=jp, and r = J(4n -p) ; 1 2m . 2n/ir j_ I ■■ ' |i(4n-y)|i(2m+j)) - EXAMPLES. XIII. b. Pases 147, 148. 1. («-2/)'°=a:'°[ l-|j . Let T^ and T^j denote consecutive terms (u\8o 30-r+l 4 1-|1 ; then T^i = . — xT„ numerically; .: T^^>T^, so 124 _4r long as — =r; >1; that is, 124 > lor; therefore r=8 makes the G"" term ll?- ' greatest. 2. (2x - ZyY^= (2^)28 (l _ ly'; ,. r^^=H8zl+i . 1^ X T„ ««m«r£. cally; :. T^-^>T^, so long as 68-2?->3r; that is, r=ll makes the 12"' term greatest. 3. (2a + 6)»=(2»)»(l + ^y'; T^^^^l^Zl+l^-^T,; .: r^i>2V, so long as 75>13r; that is, r=5 makes the G"" term greatest. 4. (3 + 2.)x»=3^=(n-|)"; T,,,=li^^fxT.; .-. Tr^i>Tr, so long as 80>8r; that is, r=10 makes the lO"" and 11* terms equal, and greater than any other term. 7-r 2 5. Tr+i>T„ so long as -o^l; that is, 14>5r. Therefore the 3'^ 6 5 4 term is the greatest, and its value = r-^.jj = 6§. 1 . ^ u f a;\" 10-r 2 6, (a+x)''=a''(H— ) ; and T^i= .yXT,; y Qi J T o •>3r. Therefore the r term. Their value rV2y V3/ 144" .'. T,.+i>2',, so long as 20-2r>3r. Therefore the i"" and S"" terms are equal and greater than any other term. Their value 9.8.7 1.2.3 62 BINOMIAL THEOREM. [CHAP. 7. We have to shew that 'i"C„ = ^'^^G^-i + 2"-^C„. |2n-l Nnw 211—1^7 ,=2"-l(7 = I — . JNOW 0„_i- "-n-|^j_i^. |2n 2n | 2m-l 2 . | 2b-1 which proves the proposition. 8. By Art. 165, we have {x + a)" =A +B, ani {x-a)"=A-B ; therefore by multiplication we get the required result. 9. We have na;'^' 2/ =240, ^5-^^' x"--^ 2/= =720, and ■ bv division, ^^-^— . - = - and ^^^^ . - = 3. Prom these two equations we ■' 3 a; 2 2 a; get — '" ~ ' = ^^-^ ; and ra=5; therefore it easily follows that a; =2, y = 3. 3 2 10. (1 + 2a: - x2)*= 1 + 4 (2a; - x2) + 6 (2a; - x^ + 4 (2a; - x^ + (2a; - a;^)* = 1 + 8a; - 4x= + 24a;2 - 24a;3 + Sx* + 32a;S - 48a;« + 24x6 _ 4a;6 + 16a;* - 32a;5 + 24a;6 - 8a;' + x^ = 1 + 8a; + 20a;2 + 8a;3 - 26a;^ - 8x6 + 20x6 - Sx' + x^. 11. (3x2 _ 2ax + 3o2)3= {3x^ _ (2ax - 3a^)}^ = 27x6 - 3 . 9x* (2ax - Sa") + 3 . 3x2 (2ax - Sa^ - {2ax - Sa^f = 27x6 - 54ax6 + Sla^x* + SGa'x* - lOSa'x^ + 81a''x« - 8a%8 + 36a*x2 - 54a6x + 27a6 = 27x6 - 54ax6 + 117a2x* - liea^x^ + ina*x' - Sia^x + 27a\ 12. The r* term from the end is the (m-r + 2)"' from the beginning In and is equal to -. =-^ l x""-! a"-'+^ 13. There are 271+2 terms in all, and the (p + 2)* term from the end has 2n + 2 - (p + 2) before it ; therefore counting from the beginning it is the {2n -p + 1)"* term, which is |2»i + l / i\2ii-p |2« + 1 ^ p + l\2n-p \ x) ' ™ ^ "-^ \p + l\2n-p 14. We have ^^0^ = ^ C^+i ; therefore 2r + r + 1 = 43, or r = 14. 15. We must have ^"Og^-i = ="G,+i ; .-. 3r-l + r + l = 2re, or 2r=n. XlII.j BINOMIAL THEOREM. 63 |2tt 16. The middle term is the (n+ 1)*, which is — x\ In In 1.234 2n This may be written —^-^ — V-— — • a". lit. « ' and remembering that 2 . 4 . 6...2re=2" . In this reduces to l-3-5-(2»-l) 2»^« t 17. This is solved in the first part of Art. 176. ,, .„,, , , ,. (n+l)7i „ (n + l)n(n-l) , 18. (l + a;)"+i=l + (n+l)a: + ^ ^^^^ a:^ + ^ 1.2.S — •"'+- + (n + l)a;»+x"+i; (l + a;)»+i-l n , n(n-l) , ii;»+i ••• n+1 =" + 172 '■'^+172:3'''+ - +^r that IS .^ '—^ = V + ir'^ +0 '^ + — + , -, • n+1 "2 3 n+1 Putting x=l, we get the required result. 19. Writing Cj, t'l, Ua,... in full we obtain n n (n - 1) n (n - 1) (n - 2) =- + — ^ ' + -^^ — , '^., — ^ + ...to nterms, In n (n - 1) that is, n + (n - 1) + (n - 2) + ...to n terms. 20. Wehave — — J=l + -= . Cj n n Ci+C2_j^ 2 _?t+l Ca + C3 _j^ 3 _n + l, Co n-1 n-1' c- n-2 n-2' .•. by multiplication we get the required result. 21. As in Ex. 18, we have (l + x)"+i-l _^ c,x' c^^ ,V'+i_ n+1 "2 3 '" m + 1 Putting x=2, we easily get the required result. 22. We have (1 + a;)»= c„ + CiX + Cas" + . . . + c„x», il + x)=''«+x+i+ •■■+#' /. — (1 + x)^ = (c^^ + Ci^ + C3" + . . . + O + terms which contain x ; 64 BINOMIAL THEOREM. [CHAP. .-. Co" + Cj" + Cj'' + . . . + c„2 is equal to the term independent of a; in — (1 + x)'\ that is, the ooefBcient of x" in the expansion of (l + x)^". x" 23. (l+a!)"=c„+Ci!i;+Cja;2+...+c^''+... + c„x", also since terms equi- distant from beginning and end have the same coefScient (l + a;)''=c„ + c„_ia; + c„_2x2+ ... + c„_rX'-+ ... + Co«" Now multiply these two series together and pick out the ooeffioieint of a;"+r . then c^c,. + CjC^-^ + c^^^ + . . . + c„_, c„ is equal to the coefficient of x""*^ in the expansion of (l + x)**". EXAMPLES. XIV. a. Page 155. 1. (l + x)4=l + Jx + l^^X^ + Lki-2i^^x3+... ,11,1, 2. (l+x)f=lH-|x.lfi-^xHiM^^'-^- 1 3 3,1, \1 3. .-..^.-i.m^M^m^^- 4. (l+x=r=l + (-2)x>+^^)xHl:^^^Hri)xe+... = l-2x2+3x«-4x«+... 5. (l-3x)J=l-i3x+l^{3x)=-iii-.lii_^(3:.)3+. = 1 - a; - a;2 - - rc^ + .„ o XIV.] BINOMIAL THEOREM. 65 6. (l-3^)-4=l-i(-3x) + i-ii:^±-j:(_3x)» 1.2 ;-l)(-H(-H,.,.,.. 14 =l+x+2x^ + -^x^+... O (i+2.ri=i+(4)2.+iiK-H, .(2x)= ■ (2x 8.(..|)-=i.,.a,|.<.4ti)(|)VM>(zi)(za(|)V.,. ^(:IKJ_;Ki:!l,2,).+....i-,4.._^..+.,. 2 „ 10 , = l-x + ^x^--^x^+... 3/3 \ „ /, 2a;\^ , 3 2a; 2V2 / f2x\^ JiH(H(|)%...=i..4.4^..... 10. (i.i.)-=i.(-,g.).i4t^(^y (_4){-5)(-6)/'l V 10,5,53, + [3 A 2"j +- = l-2« + 2»''-2«'+- 11. (2 + x)-»=2-3(l + |)~' = i(l-|x+|^='-|x3+...) 12. (9 + 2x)i=9j(l + |)*=3 {l + l-^^-' + i^s^'+-) 13. (8 + 124-8t(l+^-?^)^=4(l + |a)^ = 4(l+a-ia»+|a3-...]. H.A. K. 66 BINOMIAL THEOREM. [CHAP. 15. (4a-8.)4 = (4a)i(l--) =-^(^l+- + 2-, + 2HS+-j- 13. i(±H±I:i (2.,= -14^^2V 429 7 -"16 '"^ 17. ^\^ ^ ^^ i(-2xT 11.9.7. 5.3. l.(-l)(-3)(-5)(-7) .„_ 77 = 21^10 '^ •"" "256* • ^(^_l)(^_2)...(^-9.l) 18. ^^^ ^^"^^ ^' ^(3ar 16 .13. 10.7.4.1.(-2)(-5)(-8) 1040 = 3 , I --5^ 2(-l)(-4)...(5-3r) 2.1.4...(3i--5) a;S|^ 10. (r + l)"'termof (l + 2a;) 4 [^ (-sxr ^1^.5 (2,-1)^ 11. (»■ + 1)"" term of (1 - 3a:) ^ .IMIifHh:!),., 2.5.8... (3r-l) XIV.] BINOMIAL THEOREM. 69 12. (r + l)* term of {a^-nx) 1 _1 11 \n 7 \.« / \.« / /jmV ~a Ir \a"/ _ (B + 1) (2n + l) (3m + l) ■■■ {r^.n + 1) x' 7+5.-1 4 13. 3'^i = ; . r-= X T, numerically; .: T^i^Ty, so long as 2i + ir:>15r; r = 2 makes the S"* term greatest. ■2""'" + ^ 2 14. T,+, = ;; gXlV; .'. Tr+i>rr, so loDg as 23 > 5)-; thus the 5* term is the greatest. 11 15. T^i = I XT,; . r^i>2'r, so long as 49 + 28)->32r; thus the 13* term is the greatest. x) 16. (2^ + 5j/p = (2^p(^H-g^ _ 12-)' + l 5x3 •■• ^r+i- ,. •2x8'^ ••' that is, !r,+i>r,, so long as 195 > SI)-; thus the 7"" term is the greatest. 1/. ■'r+l— ^ .gX±r- .: 2V+-i>Tri so long as 12 + 2r>5?-; thus r=4 makes the 4"" and 5* equal and greater than any other term. iS\ —n 18. {3x^ + iy^)-^ = (Sx^)-" (l + III) 15 + r — 1 4x8 .-. 2V+i= — ~ • D — m^^r numerically; that is, r,+i>Tr, so long as 448 + 32r>243r; thus the 3'^ term is the greatest. 19. ^9?= (100 -2)^ = 10(^1 -A) ^ioU-±-l(±yA(±Y-..\ \ 100 2 Viooy 2 uooy ) = 10 (1 - -01 - -00005 - -0000005) = •9899495 x 10=9-89949... . 70 BINOMIAL THEOREM. [CHAP. 20. 398 = (1000 -2)4= 10 (l-j^) "^■"■"{■^"s* 1000 "9(1006/ " ••) : 10 (1 - -0006666 - -0000004) •999333x10=9-99333. 21. yi003=(103 + 3)*=10^1+i^ 102 10^"*"- = 10+ -01 - -00001 = 10-00999. l\i 22. 4/2i00=(7^-l)i = 7ri-^,') .7Ji-:»™l!™_i,.o(K« '"^- ^^'^^ ; the m* coefficient 2ra-2_ the (re - 1*) coefficient m - 1 EXAMPLES. XIV. c. Pages 167, 168, 169. 1. (B-5x)(l + 2!i!+3x2+... + 100»99 + 10l!i;i»»+...); .-. the required coefficient = 803 - 500= - 197. 2. See Example 1, Art. 193. With the same notation the required ffi • * ^ .o A 13-14 „ 12.13 11.12 ,.„ coefficient = 4pi2 + I'p-^^ - Pio = 4 . — ^ 2 . — ^ y~ = 142. 3. 52 — f.(l+a;)-i=i(3a;2-2)(l-a!+a;2-!es+...); .-. the coefficient of a;"= 3 ( - 1)""! - 2 ( - 1)"+^ = ( - 1)"-!. 4. With the notation of Ex. 1, Art. 193, we have the coefficient of x" = 2( l)n ("+l)(" + 2) |( l)u-l. "(" + l) | ( l.)n-2 ("-l)" = (-l)»»(»='+2re+2). -i /3\-4 5. Bxpansion=(l + ^) =Q = g)^ 6. v.==«=Gr=('-i)"'- 74 BINOMIAL THEOREM. [CHAP. 7. The first series=(l-|)""= g)~"=2".|^;=2''(l-iy". 8. The first 861X68 = 7" Cl+iy=8''=4''.2»=4»^^^ "=^"(^"1) "• 9. The expression = /n\2 =!'4-?-K?)1('-i'')('4-r ■('*l-^.-')(-i-)('-i"i-) , /'81 81 1 243\ „ , t- , -^ 256'^- 10 and 11. See Ex. 3, Art. 193. 12. The expansion={(l+a:)-2}-». =(l + a:p; |2n .•. the requued coeflaoient= 13. The middle term of ( a;+ i ) is H \1\!!l' |4m ^ 2^''|2n.l.8.5...(4>t-l) 1 . 3 .5...(4»-l) |2re J2re [^ |^ 2»[k. 1 .3. 5...(2n-l) ^2„ (2» + l)(2m + 3)..(4n-l) 4"( n + ^ J (w+n ) (" + 9) •■■*o ra factors = the coefficient of as" in (1 - 4a;) -H). 14. We have (1 - x^) = (1 - a;)^ + 3x - %x^ ; .-. (l-a;3)n={(l_ic)3 + 3a;(l-a;)}''; &o. XIV.] BINOMIAL THEOREM. 75 15. ,_, V . ^j— ^„ = (l-a;){l + a3+a6 + a;9+...}, 1 + x + x^ l-x' ^ ' <■ >' and in the series every index is a multiple of 3 ; therefore in the expansion of the given expression every index is of the form 3m or 3m + 1. In the former case the ooef&cient is 1, and in the latter it is - 1. 16. (1) See Art. 191. (2) The sum of the coefficients will be independent of a, 6, and c ; if these he each equal to 1, the whole expansion is the sum required, which is therefore equal to 3^ or 6561. 17. Multiply throughout by Ira; then we have to shew that »Ci + ''C78+''C|i + ...+''C„_i=2''-i, which has been proved in Art. 174. 18. (1) Wehave (l + a;)"'=c„ + CiX + C2a;2+...+c^'-+... + c„a:", (l + a;)-i=l-x + a;2-a;3+... Multiply the two series together ; the coeflScient of x*" in the product on the right is (-1)' {c„-Ci + C3-C8+ ... + (- If c,} which must be equal to the coefficient of x' in (l + x)"~^, that is to In-l |r in-r-l (2) (l + a;)''=Co + Cia; + C2a;2+... + c„x», .-.by multiplication, (l + x)"(l+- j =a series of terms in which the coef- ficient of x" is c„-2ci + 3c2-4c3+... + (-l)"(ra+l) e„. This expression is therefore equal to the coefficient of x" in x^ (1 + x)"-^^ that is it is equal to zero. (3) (l + x)''=C„ + CiX + C2x2+... + C„X'», + «J_... + (-1)«£E; ('-»"- .-. by multipHoation,(H-a;)''^l-iy={co=-Ci2+C2«-... + (-l)"c„2} together with terms involving x. Hence Cj'' - Cj^ + Ca" - . . . + ( - 1)" cj is equal to the term independent of x ^^"(l-xr x" This term is when n is odd, and ( - 1)" c„ when n is even, since in the latter case we have only to consider the coefficient of x" in (1 -.t^)". 76 BINOMIAL THEOREM. [CHAPS. 19. (1) (l-x)-5=l + 3x+|^:!:2+|^x8+... 1.2 2.3 3.4 „ 4.5 , n(n+l) Biuoe s„= -i-g — '- . (2) (l-a)-s=Si + s2a; + S3a;2+...+s2„s''»-i+... (1 - a;)"' = Si + 820; + Sjo;'' + . . . + S2„a;^"~' + . . . Multiply the two series together, and take the coefficient of a;^"~' ; thus *i*2?i+*2*2n-i+-" *° 2ji terms = the coefficient of x™~^ in the expansion of (l-x)-«, &c. 20. (1) It wiU be found that (l-x)~^=l + q^x + q^^+qsa?+...+q^^^x^^-»- + ... (1 -a;ri=l + 3ia; + 22a;=+?3^'+ ... + 32^ia;2"+i+ ... .-.by multiplying these results together we see that 92n+i + 2i92n+22?2n-i + "- *» 2n + 2 terms=the coefficient of a;^" in (l-a;)-', which is unity; •■• ?2j^fl + 2l22n + ?2!?2n-l+ — +3n-lSn+2 + 97i2,t+-l=2- (2) Wehave (l-a;)~i=l + gia; + g2a;2+... + g2„a;2"+...; , and (l + xY'i=l-q-^x + q^^- ... + q^x-^''~ ...; lie'ice (1 - a:2)-i= {q^ - q^q^^_^ + q^q^^_^ - ... + ffjn} x^, together with terms containing other powers of x. Now the series in brackets consists of 2n+l terms, those equidistant from the beginning and end being equal ; ••• 2 {32„ - 2i22„-i + 32S2n-2 - ... to n terms} + ( - 1)" q,? = the coefficient of x^" in (l-x^) 2 = g'„. Transpose and we get the required result. 21. We have (Co + Ci + C2+...+cJ2_2(c„Ci + C„C2+...+CiC3+...) = Co2 + Ci2 + C22+...+C„=; .: 2(c„Ci + CoC2+... + CiC2+...)=22»-.j=-, See Ex. 22, XIII. b. 22. (7 + 4 v'S)" =J) + ^, where j3 < 1. Also (7 + 4,y3)''+(7-4;^3)''=integer, and (7 - 4 ^^S)" is positive and less thanl; .-. (7 -4 ^3)" must be equal to 1-/3. Now (7+4;^3)''x(7-4V3)»=l; :. (p + p)(l-^) = l. XIV, XV.] MULTINOMIAL THEOREM. 77 23. LetS„=c,-| + |-...+izl)!:^™ £6 n =n- ' g + g 3 - ... to n terms. ^n+i=(» + l)- 2i2 + ^ 3fi ---^0 ra + 1 terma. . o o 1 n n(n-\) ^ , . •• ^n+i-'S„=l- — +— i.-5 — i-. .. to re + 1 terms = -^ {1 - (1 - l)''+i} = ^ ; .: /Sa-/S'i=5; but 51 = 1, thus S, = l + ,^, 1 1 1 '3 '^2'^S'' EXAMPLES. XV. Pages 173, 174 1. As in Art. 194 the term involving a'h^c^d is .„ '--, ., a'i-bfi-cyd; that is, -12600a2iVd. 18 2. The term involving a^b^d is — a^b' ( - d) ; hence the coefficient is -168. 17 3. The term involving a^bh is 7—5=7^ (2a)s 6' (So) ; hence the coefficient lillli IS 3360. 19 4. The term involving xh/^z* is p^ — (ax)''(-62/)'(y(~x* + 3x^-2x + l)^ + 10{-x^f{-x* + 3x^-2x + lf + ... The term containing x^ arises from 10(-x^)^(-x*)''; hence the coefficient is -10. (-\)H-^y{-i-'*^) 10. General term= i r^ ( _ 2)^ (3)1' a!^+^^' where p+2y=5, p=p+y. Thus 7=2, /3=1, p = 3; 7=1, j3=3, p=i; 7=0, j3=5, p = 5; XV.J MULTINOMIAL THEOREM. 79 .". the coefficient J-i)(-l)(-i) ^_^^,^3^,^ (-^)(-i)(-i)(-i) ^_ If. 11 . [~2)\~^)[-2)[~2)[~2) , „,5_135 105 ^63_ 3 11. General term=il^ — i—^ i i-2f(S)y{-ifa^+^y+^^, where /3+27 + 3S=3, ^=j3+7+5. ThusS=l, 7=0, /3=0,p = l; 3=0,7=1, /3=1,^ = 2; 3=0,7=0, ,8=3, j)=3; .: the coeffieient=i{-4)+i(-i) (-2) (3) +l2ll_|li-£^ (-2)' „ 3 1 12. This is equivalent to finding the coefficient of x* in the expansion of where /3+27=4, jJ=^ + 7. Thus 7=2, /3=0,i) = 2; 7=1, /3=2, p=3; 7=0, /3=4, 7=4; .....o».«.u...tMza(i)M.ai^(-|)-g) (-2) (-3) (-4) (-5) / ly 1 4 5 _ 4 ■^ |£ V 3/ ^? 27 "^81" 81' 13. {2-ix + 3x^)-'=^(l-2x+^xA~''. f 3 X"'' The general term of the expansion of ( 1 - 21 + - x* J is <-^n-3).^-(-2-l' + l)(_2)P ('|yxP+2v, where ^ + 27=4, i,=|3 + 7. Thus 7=2, j8=0,2)=2; 7=1, /3=2, i)=8; 7=0, j3=4, p = 4; 80 MULTINOMIAL THEOREM. [CHAP, .-. thecoefflcient=(^(|)%b2)(^(_2)=(|) (-2)(-3)(-4)(-5) ^^^^^_^,^3^^59_ 14 ' ' 4 4 59 Thus the coefficient required==-^. lb 14. This is eijuivalent to finding the coefficient of x' in the expansion of (1 + 4a; + 10x2+20x3)-*. General tenn=\ ^^ ^ f/""',\ ^ / {if (10)T {20)» x^+2y+3S Here (3+27+35=3, p=j3 + 7 + S. Thus «=1,7=0,|8=0, j) = l; 5=0,7=1, /3=1,^ = 2; 6=0,7=0, j3=3,i)=3; ••■ tl^e ooeffioient= ( - 1) (20) + ^ - 1) ^ - ^^ (4) (10) + (i)iJK:ll(4)3=_i5 + 15!. [3 ^^' -^^"^2 2~ ^• 15. This is equivalent to finding the coefficient of x* in the expansion of (3 - 15x + 18x2)-i, or i (1 - 5x + 6x^)-K General term= ^~^) ^~^^ •"^"^~-P + ^ ) {-5f (6)Yx^+2v Here j3 + 27=4, p=^+y. Thus 7=2, ^=0,i)=2; 7=1, ,8=2, j,=3; 7=0, ,8=4, p=4; .-. the coefficient =i^^i||^) (6)2 + (-^(--^)(-3) ( _ 5)2 (gj ■f ^-^^<-^}i-^'<-^' (-5)^=B6-450 + 625 = 211. 211 Thus the coefficient required = -5- , o 16. (l-2z-2x2)i=l-i(2x+2s2) + iiLi^(2x + 2x2)2... = l-sa;-3x''-|x='=l-ix-|x2. X.V.] MULTINOMIAL THEOREM. 81 17. (1 + 3X2 - 6a:3)-| = 1 _ ? (3a;2 _ 6a;3) + \ V \ ^' (3^2 - 6x3)2 + = 1-2x2 + 4x8 + - (9x*- 36x5+. ..) + ... = l-2x2+4xS + 5x<-20x'. 18 (8-9x' + 18x4)^=16(l-|x3+|x«y fi 9 3,9 A^ 1 4/9 3 9^^ , WAli/9 , 9 ,y = l-|xs + 3x« + ^(x3-2xy + ... = l-|x3 + 3x* + ^x8-|x7 + |x8; and the required expansion is obtained by multiplying this result by 16. 19. The first part is obtained by putting x = 1, for then l+x+x' + ...+x»=j) + l. For the second part, change x into 1 + x ; thus a„ + ai(l + x) + 02{l+«f+a3(l + x)3 + ...+a„j,(l + x)"i> = {l + (l + x) + (l + x)2+(l + x)3+...+(l + x)!'}'' = {1 + 1 + 1 + .. .to (p + l) terms+ (1 + 2 + 3+.. .to i) terms) X + higher powers of x}" = U + l+^-i^ — ^x+higher powers of x| =(^ + 1)" f 1+^+ ... j = (^ + 1)" (l + -^x+ higher powers of x ) . Hence by equating the coefficients of x Oi + 2aa + Soj + . . . + mpa„p = ^ ?ip (y + 1)". 20. In the expansion of (l + x + x")", since the coefficient of x' is unity, it is evident that the coefficients of the terms equidistant from the beginning and end are equal ; hence (l + x + x2)"=ao + aiX + a2X='+a3x'+... + o„x''+... + Ojx^'-s + a^x^n-a + a^ajSn-i + af,x^. Writing - x for x, (1 -x + x2)''=Oj-aiX + 02x2- 03x8+... + (-l)"a„x''+... - Ogx^^-s + a^^-^ - aix2»-i + a„x^. H. A. K. 6 82 LOGARITHMS. [CHAP. Multiply together the two series on the right; the coefficient of x^ is o„2-< + 022-a32 + ... + (-l)'»-iaVi + (-l)"a»"+(-l)"-'aVi+- or 2{a|,2-ai= + 02!'-03='+... + (-l)"-^aVi} + (-l)''V; ^^^ it is equal to the coefficient of x^ in (l + aj + x^)™ {l-x + x^ °^ (l + x^+x*)^; that is, it is egual to a„. Hence the result. 21. We have {l + x+x^)^=ao+aiX+a^''+a^^ + (1). Denote the cube roots of unity by 1, u, a^. By changing x into ax, a'x successively, we have {l + wx + a''x^)'^=aQ + a-jiax + a2x*+ (2), (l + i^x + oix^)''=a^ + aj(iiH + a2Wx'' + asX^ + aiUr'x*+ (3). Put x=l in (1), (2), (3) and add the results; then since l + a+(ir'=:0, we have 3'^=3{afi + a^+ag + ...); whence we have the first part of the question. Multiply (1), (2), (8) by 1, u^, a respectively, put x=l, and add the results; .-. 8'>=3 (01 + 04+0,+ ...), which is the second part of the question. , Finally, by multiplying (1), (2), (3) by 1, w, a^ respectively, putting a; = l and adding, we obtain the last part of the question. EXAMPLES. XVI. a. Page 178. Examples 1 to 14 are too easy to require full solution; the following six solutions will suffice. 1. (2) Let X be the required logarithm, then (2;^3)'^=1728 = 123=(4.3)'=(2^3)6; .-. x = fi. 2. (2) (4)»:= -25=1=4-1; .-. a;=-l, 5. (2) (9^/3)-=-i=i=3-^; .-. (32.34)=^= 3-2; 3t^=3-='; .: x=-^. 8. log(Va^)'=log(o6.69) = 61ogo + 91ogl>. 9. log{^^ X ^6') =log(olx6^)=|logo + |log5. =logc-°= -Slogc. XVI.] LOGARITHMS. 83 =jlog5 + j^log2-iaog3'+Iog2)-llog2 = llog5-(l + ^-^)log2-|log3 =Jlog5-2log2-|log3. 16. log 4/729 4/9-1 . 27^=log (36 . 3-t . 3-*)i=log (36 . 3-'')i=log3. ,„ , 75 „, 5 , 32 , (75 32 /9Y} , „ 17. log_-21ogg+log2j3=log|jgX2^x(gj |=log2. 19. Taking logarithms we have 2xloga + 3a!log6 = 51ogc; .'. a;(21oga+31og6) = 51ogc; 51ogc ~2 loga + 31og6" 21. Here 2a;loga+32/log6 = 51ogm (1), 3a; log a + 2jf log 6 =10 login (2). Multiply (2) by 3, and (1) by 2 and subtract, then 5a; log a=20 logm; .■.x=!i?i^,and2,= -^. log a ' logo 22. We have 2 log a; + 3 log 2/ = o, and log a; -log 2/= 6. Whence loga;=v(a + 36), log 3/=-=(ffl-26). o o 23. Wehave62i=aa;+5-<3-«. _._ ix^gx+i. .-. a;log6 = (a; + l)loga; that is, a; (log 6 -log a) = log a, or a;log ( - j=loga. 24. We have (a^ - b^-jux-a =(a-b)^{a + Vf^ ; ''• (a + 6)-2 ~(a-6p-2' that is, (o + 6)2*=(a-&)2; 1 , ,11/ n log (a- 8) .-. a;log(a+5)=log(a-J),a:=j^^^j. EXAMPLES. XVI. b. Page 185. 2^ 6—2 7. log •128=log 1^=7 log 2 - 3=2-10721 - 3 = 1-10721. 4l LOGARITHMS. [CHAP. 13. log (-0105)^= jlog (5ii?iil) =i (log5 + log3+log7-4) =i (2-0211893)= J (I +2-0211893) = 1-6052973. 14. log 324=21og2 + 41og 3=2-5105452; .•. if X be fjie required 7* root, we have logx=ilog (-00324)=^ (3-5105452) = 1 (7 + 4-5105452) = 1-6443686 ; .-. a; =-44092888. „ , 2 , 392 2 , 72x2' 15. iog^=ni°gio=n'°s-io- = ^ (2 log 7 + 8 log 2 - 1) = -28968836 ; .-. a; =1-948445. 16. Let P be the product ; then log P= sum of logs of its factors = 1-5705780+ -5705780 + 3-5705780 + 5-705780 = 5-2823120; .-. P=191563-l. 8^ . 5^ 2 4 1 17. logx=log — 5— = 5log3+5log5--log2 = '^^^^ + 1 (-69897) - I (-30103) = 1 -1998692. 18. logx=log72«73 + log^P72a-logi!y273 =3log2 + 2log2-^log2 + |log3 + |log3-ilog3 =jlog2+log3 = l-0039288. 19. logx=log(^ ^^g^^, j .-n K (log 7 + 3 log 5 - 5 log 2) = -9579053 ; , x=9-076226. XVI.] LOGARITHMS. 85 20. log a: =^?^y- log (22x70)4 = (41og3+logll + l-21og7)-i(log2 + logll+log7 + l) =41og3 + H(i+iogll)-^log7-ilog2 = 9-3935917-7 -1428266 = 2-2507651 ; .-. a; = 178-141516. 21. log a; = 12 log 3 + 8 log 2 = 5-7254556 + 2-4082400= 81336956 ; .'. X contains 9 cligits. on o- /21\"» /3x7\i«» 22. Smce y =i^^^^ ; •■■ ^°^ ( 20 ) = -^^^ ^^°^ 3 + log 7 - 2 log 2 - 1 + log 2) = 132-22193 - 130-10300 = 2-11893 ; /2l\ioo ■'■ I an ) ^^ * number which has 3 integral digits; that is it is greater than 100. \1000 ~) =1000 (-log 2) =-301-08 =802 -97; .-. there are 301 ciphers between the decimal point and the first significant digit. =3-46... 24. 3'«-2=5; .-. (a;-2)log3=log5; log 5 + 2 log 3 _ 1-6532126 •'• '^~ log 3 " -4771213 25. a;log5=31oglO, a;=;g^=4-29... 26. (5-3x)log5 = (a: + 2)log2; or (5 - 8x) (1 - log 2) = (a; + 2) log 2, a;(log2 + 3-31og2) = 5-71og2; .-. a; (3-2Iog 2) = 5-71og2; 2-89279 , „„„ •■•^ = 2-^39794 = ^''°^- 27. Here a; log 21 = (2a; + 1) log 2 + a; log 5, a: (log 3 + log 7) = 2a; log 2 + log 2 + a; - a; log 2, a;(log3 + log7-log2-l)=log2; log 2 _ -30103 _.,„„(. •■ '^~log3+log7 -log 2-1 ~ -0211893" "•- LOGAEITHMS. [CHAP. 28. We have 2». 6»^-==52»:. 71-"^, a;log2 + (a;-2)(Iog2+log3) = 2a;(l-log2) + (l-a;)log7; .-. a!(41og2 + log3+log7-2) = 21og3 + 21og2+log7; 24014006 5263393 =4-562... 21-11 = 6'' ) 29. We have ^^ =3. 2^+4 ' .-. (a;+jr)log2=2/(log2 + log3), a;log3=log3 + (2/ + l)log2; thatis, a; log 2 =2/ log 3, a;log3-ylog2=log2 + log3; .-. by substitution, x { {log 3)^ - (log 2)2} = (log 2 + log 3) log 3 ; log 3 , log 2 • " ;, and 2/=-- -^ log3-log2' " log3-log2* 30. Put log 3 = a, log 2 = 5 ; then we have ax + (a-2b)y-a = 0, {2b+a)x-Say-b=0; .: by cross multiplication, X y 1 b(2b-a)-Sa> -a{2b + a) + ab -Sa'-{a-2b){2b + a}' X « 1 or . ^ £ ^ . 262_a6-3a3 -(ab + a^) Hb^-a^)' thatis, (6 + o)(26-3a) -a{b + a) 4(6 + a)(6-a)' _ 2&-3a _ 31og3-21og2 _ a _ log3 •■ '"~4(6-a)~4(log3-log2)' *"" ^ " il^Tj) " 4 (log 3 - " -log2)' 31, Let a; = log^j 200, then 25"^ = 200, 2a;log5=2+log2, '^=§^2=1-6465. 32. Let x=log,^2, then T'=^2, 1 rr 1, o -150515 ,„„, a.log7=2log2, .-. a.=.^^g^ = -1781. Again, 2i*=7; hence 5a;log2=log7, a3iix=%^^=5-GU. ^ log 2 Orthus, log,^2xlog^27=l; -■- log^27=j^ = :jL_ = 5-614. XVII.] EXPONENTIAL AND LOGAEITHMIC SERIES. 87 EXAMPLES. XVII. Pages 195—197. >^3 ^3 Q.i 1. In the equation log, (l+a;)=a;--^+ ^ --j- + ... ^ D 4 puta;=l;then l-^ + ^ - j+...=log,2. 2. We have ^ _ ^-1-, + _!_. __!_+... =IogJl + 2J=log,-=log,3-Iog,2. 3. log.(jn-o)-log, (n-a)=log, U ri+^j| -log. |n Tl-^J | =log,7i+log, f 1+lj -log.ra-log, f 1-^j =Iog.(n-^)-log;(l-g; and the result foUows by using the formulse for log, (1 + x), and log, (1 - a:). 4. We have y=log,{l + x), hence l + x=e''=l + y + ^ + f^ + ... 5. The series on the left = - log, {'-"-i-') = - log, - =log, a - log, b. a 6. In the result of Example 3, put n= 1000, a=l; ••■ 1°8. 1001 - log, 999=2 ( ji^ + J . jA^ + 1 . jlg^,) ; the term = . ^ ^ and subsequent terms may be omitted. ^ , x2 a;2 7. In the series e*=l+a! + -^ + 7^ + ..., put x= - 1, then , „ ,, /l A /l 1\ 3-1 , 5-1 , 7-1 .-i=(l-l) + (^-^j + (|^-^) + ... = -j3- + -j5- + ^+. which gives the result. 88 EXPONENTIAL AND LOGARITHMIC SERIES. [CHAP. 8. log. {l + xy+''{l-xy-'^={l + x) log.{l+a:) + (l-a;) log, {1-x) = X {log, (1 + x) - log, (1 - a;) } + {log, (1 + x) + log, (1-x)} whence the result. 9. x^-y^+-^{x*-y*) + hx^-y') + ... 10. Put 71= 50 in the formula The right side =-00868589 + -00008686 + -00000116 + •OOOOOOOl = -00877391, thus 2 -log 2 -21og 7 = -00877390; whence log 7 is found. Put m=10 in the formula log,„(™+l)-log„n = ^-J^ + J^3-... Positive terms. •04342945 14476 87 Negative terms. •00217147 1086 7 1 -04357509 218240 •00218240 -04139269 Thus log 11 = 1-0413927. In the last formula put k=1000; .-. logio 1001 - logio 1000 = -0004343 - -0000002 ; .-. logio (7 x-llx 13) = 3-0004341, and logn, 13 =3-0004341 -log 7 -log 11. 1-, mi. - ^ X* x^ 111 11. The expression = ;.=>-_+_-... + --_ + __... = log(l + ic2) + logri + i) = log (1 + x^) (l + i)=log {^ + ^'+^^ . 12. log,(l + 3x + 2a;2)=log,(l + a;) (l + 2a;)=log,(l + a;)+log,(l + 2x) / x'' x3 x4 \ /„ 4x2 8a.3 i6a.4 v = (^-2+¥-T+-) + (2x-^+--^ + ...j; whence the result. XVII.J EXPONENTIAL AND LOGARITHMIC SERIES. The general term of the series is r r * ' ■ r ' 13, The expression = log, (1 + 3x) - log, (1 - 2x) /„ 9x3 21x0 81x4 \ /„ 4x2 , 8x' , 16x« , N whence the result. The general term is i '- ^ — - + - — - or ^ '- x^ 6 r r r 15. e"+«-"==(i+i^+-|r+-|3;+-) + i^-'^+]2;-]r+-j „/, i2x2 i*x4 i'xs \ t2 .r^ .16 + ... 16. log, (x + 2h) + log, X - 2 log (x + fc) =log. -^^q:^' .-. the expression=l- To + Ti ~ Tr " «, / HI x(x + 2S , X - 2 log (x + fc) =log. -^^-j-^ =^°^'r"(^+ft)'4"~!(x + ft)-''"'"2(x + ft)4"^"T 17. o+j3=i), oj3=g; .-. log, (1 +i)x + 2x2) =iog, {1 + (a + /3) X + a^x^} =log. (1 + ax) (1 + /3x) = log, (1 + ax) + log. (1 + /3x) ; &o. 18. S=(l-l)x+(l-i)x2+(l-J)x' + (l-j)x4 + ... =x + x2+x3 + x4+... + f -x-2x2--x3--x^-...j 19. log. (l+^)"=« l°g.^^= -"l°gTO= -"^°^' (^-^1) = -(n + l)log.(l-^-i^)+log,(l-^) . 90 EXPONENTIAL ANB LOGABITHMIC SEEIES. [CHAPS. Hence by putting k=n + l, we have l,^ 2j k \2 3j Ifi \% ijk^ ••■ Unless re is a multiple of 4, tlie term inYolving a" comes only from ;, (1-a), and its coefficient then the coefficient of x*^ is log, (1-a), and its coefficient is — . If ra is a multiple of 4, putm=4 4m m ~ 4m ~ n ' 21. Thegeneralterm=!^ = r^=l±iM±M(!Lzi) Ira [ra-1 |m-l 1 3 \n-X \n~2 \n-3' Thusthegivenseries=l + (l + 3)+(^ + A + i) + (^^ + |. + ^ + ... 22. The result follows at once from Art. 224 by subtracting (1) from (2). = -log n , m + 1 , /, 1\ 1 1 1 24. We have (omitting the base) log2-21og3 + log5 = a; 81og2 + logB-21og5 = 6; - 4 log 2 +4 log 8- log 5 = c. Solving these simultaneous equations, we obtain log2=7a-26 + 3c; log3 = lla-S5 + 5c; log 5= 16a -45 + 7c. XVII., XVIII.] INTEREST AND ANNUITIES. 91 Nowa=-log,A=-log,(l-l)=l+^ + ^, + ... = .105360516. c=log. (l + ^) =^ - ^^ + _1^^ _ ... = -012422520. EXAMPLES, XVm. a. Page 202. 1. We have Jlf =100 (|^ j . .-. Iogj;f=2 + 50{log21-log20) = 2 + 50(log3 + log7-l-log2) = 2 + 1-059465; .-. H= 1146-74. 2. We haya Pjir=90, and -r— — = 80. Substituting for nr, we ottain ^" ==80; whence P=720. P+90 /21\'» 3, Let n years be the time, then P (K(;j =2P; whence Ji(log21-log.20)=log2, and ™=||nl^=14'2- 4, r = 10000 (|^y^ whence logF=4-(8x •0211893)=3-8304856; that is, F= 6768 -394. 5. Here 2500=1000 fjgj ; that is, 10=4 fj^j ; "3979400 whence 1 = 2 log 2+» (log 11-1); thus ji=;^jjgg2^=9-6. 6. We have D = j^ , and Z= Pnr. 1 _ l + «r __l_,l_i,= where H is the harmonic mean be- •■ D~ Pnr ~ Pnr^ P IP -^ TT tween I and P; thus X>= y /21\ioo 7. We have M=P l^j ■ .-. log af= log P + 100 (log 21 - log 20) = log P -f- 2-11893 ; .-. M=P X 10211893 ; that is, M is greater than P x 100. 8 The sum is the present worth of £1000; hence F=1000(l-06)-i2; .-. log F= 3 - (12 X -0253059) = 2-6963292. Thus F=496-97. 92 INTEREST AND ANNXHTIES. [CHAP. 9. If n is the number of half-years, then 6000=600(1-18)"; or 10=(1-18)''; .-. 1=m log 1-18; m=;g^jgg2=13-9. 10. if =(l-06)2»o farthings, logJlf=200x -0253059 = 5'06118; Jlf= 115027 farthings. EXAMPLES. 2VIII. b. Page 207. 1. In Art. 237, put A = 120, n= 5 ; then 672 = 600 + 10 x 120r, whence Ifa;=(l-045)2», logx=201ogl-045=-882326; .-. x=2-4117; .^^20000x1-4117^3^3^^^ 3. Here £2750 is the present value of a perpetual annuity of amount A say, interest being reckoned at 4 p. c. Hence by putting r = ^^ in Art. 240, we have 2750=25^, and ^ = 110. 1 9ft 1 90 4. Here 4000 = — , whence lOOr =~=S. r 40 5. By Art. 241, the number of years' purchase =-5^ = 28f. 6. The rate per cent, is 4 ; hence the amount at the end of two years 7. The rate of interest is 5 per cent. ; let A be the annuity; then 2522 =4 g^^^"/ = 20.4 |l - (^gyi ; [Art. 240.] 20'"'^ whence 2522 = 204 x ^^^ or ^ =^ = 926^. 8. This is equivalent to finding the present value of a perpetuity of £400 to commence after 10 years. Hence by Art. 242, F=i5^2iM:^=10000x (1-04)10; .-. logF=4- -170333 = 3-829667; and F=6755-65. XVm.] INTEREST AND ANNUITIES. 93 9. Let P be the sum, then using the formula M=Pe"'", we have 500=Pc, or P =500e-i= 500 X -3678. 10. Equating the present value "^ ^^~ -^ ' [found in Art. 240] to mA, we have m=- — iS-1 ■ Hence we have 25 = — r——, and 30= " ; whence by division l + iJ-"=^ = ^,andJS-'' = i. 5 4 4. ••• 25= —-, whence B-l = — , that is, t=—^, and 100r=3J. 11. Let A be the number of pounds paid annually, then £5000 is the present value of an annuity to commence at once and to run 10 years • ■ 5000-.l <^-^""^"'°) -^- ^"" ..5000-4 .^^ , 4-j--pjj^3j-„. Now log (l-04)-i<'=- -170333 =r829667; .-. 1-04-" = -676031. 1800 f 1 — B"") 12. The present value of an annuity of £1800 is ^ — = ; ; and the XI — 1 man will be ruined if this is greater than £20000. Now P-l=5j;; thus 20 4 9 he will be ruined if 9(l-iJ-")>5; that is if i2-»<-, or if iJ»>-, when B = 17. 21 Now log JBi7=171og 2^=17 (log 7 + log3-l-log2); and log|=21og3-21og2. By comparing the values of these expressions we arrive at the required result. 13. Thefine=^{(l-06)-i3-(l-06)-=i"} = ^(4-P),say. Now log4=-18x-0253059=--3289767=l'6710233; thus 4 = -4688385. log £ = - 20 X -0253059 = - -5061180 = 1-4938820 ; thus B = -3118042. .-. the fine=.5^x ■1570343 = 1308-619. 94 INTEREST AND ANNUITIES. [CHAPS. 14. As m Example 10 we have 1-JJ-" ^ l-iJ-2» 1-JS-S" ''=-B3r' *=b:o-' ''=^3r- .-. l + B-"=-, andl + iJ-"+B-=" = - a a \a J \a J a a a' a 15, The present value of £10 due 1 year hence =£jpjr^, £20 ... 2years = £ yY:j^^ , and so on ; 20 ' (1-05)' .•. the present value of the annuity in pounds 10 20 30 40 ^ a 1 A <, 1-05 (1-05)2 ' (1-05)3^ (1-05)* a _ 10 X (1 -OS)" _ 10x1-05, ^J^ere «=i^. andx = ^; .-. present value in pounds = ^^ _ ^^, - ^.^^ ^ .^^^^ - .^^^^ , .-. present value =£4200. EXAMPLES. XIX. a. Pages 213, 214. 1. Multiply together the two inequalities, db+xy>2 Jabxy, and ax + hy>2 ijaxby. 2. Multiply together the three inequalities, i + oisjbc, c+a>2,Jca, a+b>-2,Jab. /■ 1 \^ 1 3. (is/x--T-) >0; thatisa; + ->2. 4. We have 2ax < a'^+x^, and 2by

i; thus a- 6 is positive, ,■. a"-' > 1"-^; or -i; > i^-; hence the result. Again, 6 (^^ . ^5^ . ^^^ji^ „, > ^^^ . Similarly, an/^+ys^+zx^^ 33^2. 8. a' - 3a62 + 263 = (a - 6) (^s + »& _ 262) = (a-6)(a-6)(a + 26) = (a-6)2(a+26); which is always positive, hence a' + 26'>3a62. 9. ai-a36-a6»+6*=(a-5)(a'-6') = (a-5)2(a2+a6 + 62), thus a*+&*>a%+aJ3. 10. 6^+c2>25c; hence (62+c2)a>2a6c. Similarly (c" + a^) 6 > 2a6c, and {a^+V^o 2oCbc. By addition, (ft^ + c^) a + (c^ + a^) 6 + (a^ + 62) 6a5c ; that is, 6c(6 + c)+ca(c + a) + a5 (a + 6)>6a6c. 11. 6' + c2>26c; hence (62+c2)a2>2a26c. Similarly (c^ + a2) 5= > 2a62<;, and (a^ + 62) gs ^ 2a6c2. By addition, (62 +c^a^+ (c* + a2) 62 + (a2 + 62) c2 > 2a26c + 2a52c + 2a6c2 ; that is 2{52c2+c2^2 + a262)>2a6c(o+6 + c). 12. a;'-a;2_a;-2=(a;-2) {a2+a; + l), which is positive or negative ac- cording as X is greater or less than 2 ; .■. a;'> or •cx^ + x+2, according as a;> or <2. 13. a^ - Sax" + ISa^x - 9a^= {x - a) {x^ -4,ax + 9a2) = (a;-a){(s-2a)2 + 5a2}. By hypothesis the first factor is positive, and the second factor is always positive; hence the result. 14. l\-Vlx + lx^-!i?=(\-x) (ll-6a;-l-a;2) = (l-a;) {2 + (3-a;)2}. The second of these factors is always positive, but the first is only positive so long as a; < 1 ; hence the greatest value of a; is 1. 15. a;2-12a;+40=(a:-6)2-i-4, and is a minimnm when x=&; its value being 4. 24a: - 8 - 9a:2 _ g _ (4 _ 31)2^ and is a maximum when 4 - 3x = ; its value being 8. 16. It is easily seen that r (n - r + 1) > n ; thus we have l.n=n 2(ra-l)>n 3(ra-2)>7j (n-2)2>ra n. X=n. By multiplication the required result is obtained. 96 INEQUALITIES. [CHAP. Again, since the geometric mean is less than the arithmetic mean we have the n inequalities ; 2.271<(k+1)2; 4(27i-2)<(ji + l)2; 6(2n-4)<(re+l)2; (27i-2)4<(« + l)2; 2n.2<(n+V)\ Hence by multiplication, (2.4. 6 2nf<{p,+ l)'^. 17. ByArt. 253, ^;t|i-^>(a:!/z)4. Cube each side. Q 18. The solution is similar to that of Ex. 16. I.(2n-l)2i+in-3+--Kn-« or>2 2 ; 2" — 1 ^^ hence > 2 ^ that is > J2"-i; whence the result easily follows. 20. li±^!±^±::-t!^>(13.2a.33...„3)L , !LM!>{(L»)f • Eaise each side to the ji"" power. 21. In Ex. 17 we have proved that (a + 6 + c)3>27a6c. Put a=y+z-x, b=z + x-y, c=x + y-z, so that a + i + c=x+y + z; we then obtain the result. Again (6 + c) (c + o) (a+6)>8aSc. [Ex.2.] With the same substitutions, 6 + c=2x, c+a=2y, a + 6=22; thus the result follows. 22. The expression is a maximum when \—^) ("T~) '^ * maximum. But the sum of 4 f-j^ j and 5 ( -|^) is constant. Hence the maxi- . , 7-3! 2+a! mum IS when — j— = —-- , or a;=8. 4 6 23. Put»=i^±fn^). Thenifl + »;=2,, (4 + w){l + u) 4 „ / 2 \2 XIX.] INEQUALITIES. 97 2 2 Hence u is a maximum when -- — iijy=0; that is when y=2, x=l; in this case the value of u is 9. EXAMPLES. XIX. b. Pages 218, 219: 1. We have >f — ^j . [Art. 258.] By clearing of fractions we have the result. 2. ByArt.258,^^ + ^° + ^'+••■+"^f ^+^ + ^ + ^••+" y; H \ n J ' and therefore > ( —^ \ . By clearing of fractions we have the result. 3 ByArt.258, ^"' + ''"+'''"+- + <'")"^ P + ^ + '^ + - + ^" r- n \ n J ' and therefore > (n + 1)'". By clearing of fractions we have the result. 4. 11 a>b, [1 + ^" > (l + ^\ . [Art. 259.] Put - = ~, -= -, so that a = ax, 6 = fia;: .•. (1 + -) > 1 + -| . a a p \ aj \ pj By taking the x* root we have the result; for since a>h, a must he greater than /3. Also since x may be any positive quantity, a and /3 may be any positive quantities subject to the above restriction. Thus the expression ( 1 + - ) gradually iacreases as n increases. When n = 1 its value is 2, and when n is infinite its value is e. 5. Put c=ax, and c = by, so that a = - and 5=- . Then X y o c e4:yx; hence the result follows from Art. 2G0. 6. Consider the expression a", h'', c', ... fc*. If any two of the quantities, a and 6 suppose, are unequal, this expression is diminished when we replace a and 6 by the two equal quantities ^±i. ^±i. [Art. 261.] H. A. K. 7 98 INEQUALITIES. [CHAP. Hence the least value of the expression is when all the quantities a, b, c, ... ft are equal ; in this case each is equal to — . 7. — log(l + o"»)<;-log(l + a") if nlog{l + a™)ji, therefore a" > a™, andl + a" > 1 + a" ; hence a fortiori (l + a")'»'>(l + a™)". / 1 \™ / 1 N" Dividing this inequality by a™", vre have I— + 11 >(-^+l) • Ifa>l, then -<1, and the inequality stiU holds. 8. -T— ::s- = l+ , ....„ =1 + 1 =^+r-T-T r- X X X^ iB" Since a;{^\. [Art. 257.] But — jr— is the arithmetic mean of a and c, and is consequently greater than 6 the harmonic mean. [Art. 65.] Hence a /ortiori a" + c''> 26". \] ( — S — ) This expression is the product of 8 factors whose sum is 3 ( q ) + 5 ( — ? — ) or 4a, which is constant. X ^Qi — X 3 I— Henoe the maximum value is when - = — = — , or x=-. Thus the maxi- o ^ 33 . 5^ mum value is — ^^^a"- The second expression is a maximum when its sixth power is a maxi- mum ; that is when x' (1 - a;)^ is a maximum. As in the preceding case, this is when ^ = -^—, or a; = - . Thus the value required is . /= . // v . XIX.] INEQUALITIES. 99 11. log (1 + is) < a; if (1 + a;) < e* ; this is obviously the case since e"'=l+x + T;5 + if. Again log(l + !c)>, ,U l + x>e^+''. i + x Now l + .=_-i^=l + ^ + _|L.^H._|L. + ... 1 + x 12. Consider the expression - H (- - , and suppose z constant, so that X y z iu „ j; , • 1 X i nT 1 1 "' + '!/ constant , „ the sum of x + v is also constant. Now - + - = — - = , and the X y xy xy denominator is greatest when x=y; thus - + - is least when x=y. Hence X y if any two of the quantities x, y, z are unequal, the expression - H (- - can X y z be diminished, and its value is a minimum when x=y=^z=^ . o Thus the minimum value of - H (- - is 9, X y z Clearing of fractions, yz+zx + xy> 9xyz ; and 1 - (a; + j; + z) = ; .-. 1- {x + y +z) + {yz + zx + xy) - xyz>Sxyz ; that is, (1 - x) (1 ~y)(l-z)> Sxyz. 13. The expression (a + 6 + c + d) (o^ + 6^ + c^ + d^) - (a" + b^ + c^ + cP)'' = a1)(a-bf + ac{a-c)^ + ad(a-d)^ + bc{b-cf + bd(b-df + cd{c-d)^, and is therefore positive. 14. Since both of the expressions involve the letters a, b, c symmetri- cally, we may suppose that a, b, c are in order of magnitude; let us suppose then that a > 6 > c. In this case c{c — a) (c - 6) is positive. Also a{a-b)ia-c) + b(b-c){b~a] = {a-b) {a^ - ac - (b^ - be)} = (a-6)2(a + 6-c), and is therefore positive. Again c^ (c -a){c- b) is positive. Also a2 (a - 6) (a - c) + 62 (6 - c) (6 - a) = (a - 6) {a3 - a^c - (63 - 52c)} = (a-6)2(a2 + a6 + 62-ac-6c); which is positive since a?-ac, and 6^*- 6e are positive. 7—2 100 INEQUALITIES. [CHAPS. 15. In Example 7 we have proved that if m>m, (1 + a'')"'> (1 + o™)". Put a=- and multiply both sides by a;™", thus (a;"+2/")"'> (a"" + j/™)". 16. Let P^il + xy-'il-x)-^-^; log P = (1 - k) log (1 + a;) + (1 + k) log (1 - a;) = {log (I + k) +log (l-as)} -xjlog (l + s) - log(l-a;)} „/'a;2 x^ x^ \ „ ^ x^ x^ \ .-. log P is negative ; /. P < 1 ; .-. (1 + x)^-^ (1 - x)^+^ < 1. Now proceed exactly as in Art. 2G1. 17. Let the three quantities p, g, r be in descending order of magnitude. Then the given expression =a^{p-q){p~r)-'b^{p~q){q-r) + c^{p-r){q-r), and win consequently be least when the second term is greatest ; this is when b = a + c, which is the extreme case when the triangle becomes a straight line. The expression then = a^{p-q){p-r)-{a'^ + 2ac + c^){p-q){q-r) + c^{p-r){q-r) = a^{p-q)^- 2ac (p -q){q-r) + c' {q - r)^ = {« (p - g) - c (g - r)}2, which is positive. Hence the expression is always positive. (2 ) Substituting 2 = — (a; + ?/), we ha ve to shew that — (a^y + 6^a;) (a; + j/) + c'xy must be negative ; that is (changing the signs) we must prove that Wx'' + a%/^ + (a^ + 6'' — c^) xy is positive. This expression is equal to ^x-ayf+\[a,-^'bf-(?\xy, which is positive ; for a + 6 > c. 18. Lemma. If a + J = n, a given quantity, then la 16 becomes less and less the nearer a and 6 are to each other. For |w-r \r > I n - (r + 1) |r + l ,ifn-r>r + l; that is if n>2r + 1. Hence |m-l |1 > |w-2 |2> \n-3 |3, Thus if a+6=2m, the least value of la 16 is \m Im ; and if a + 6 = 2m+l, the least value of la [5 is Im + l |m . By the preceding lemma, |2re-l |l>ln |n, |2ra-3 |3> \n \n , |2re-5 |5> |TO_|n, , |3 |2k-3> |ra [m, 11 |2m-l > |re [n. Multiplying together these n inequalities, we have ([1|3|5 \2n-lY>(\ny-\ XIX, XX.] LIMITING VALUES. 101 19. Consider the expression [a^|6 Ic Id ; then if any two of the quantities, a and 6 say, are unequal, we can without altering their sum diminish [a |6^ by taking a and 6 equal. [This is proved in the lemma preceding Example 18.] Hence the value of la 16 Ic \d is least when all the quantities a, 6, c, d, ... are equal. If however n is not exactly divisible hyp this will not be the case; suppose then that q is the quotient and r the remainder when M is divided byp; thus n=pq + r={p-r)q + r{q + l). Hence p-r of the quantities a, 6, c, d... will be equal to q, and the remaining r will be equal to g + 1 ; thus the least value of the expression is ( W )""''( I ? + 1 )''• EXAMPLES. XX. Page 228. , ,_,-.. ., {2x){-5x) 10 ,„, _. .^ (-3)(3) 9 1. (1) Lmiit=: ^ ';^^ ' = --^. (2) Limit = ^ i l' 2. (1) Limit = ^^ = 9. (2) Limit = ^-^ = ^ . 3. (l)Limit = g^4. (2)Limit=i:^ = |. 4. (l)Limit = ^.^g^=-M. (2)Limit==lf|^l = G. 5. (1) Limit= 2^ X — = 1- (2) Limit=— ^ x ^ = 0. 6. (1) Limit=^^^^ = l = 0. (2)Limit = ^-^3=-30. x^ + l _ x^-x + l _ 3 3 '• x'-l~ x-1 ~-2~ 2' d'-l'' 1 L , x2(loga)2 , , ,^ a;2(log6)2 ) -^ = -|l + xlog 1, divergent. If a; = l, the series becomes the vergent. and the sum of the first n terms is 1 =^ . Hence the series is con- m + 1 a;" ^ x"^i {2n-3){2n-2) M„_i (2re-l)2n ' (2n - 3) (2ji - 2) (2m-l)27i If a; < 1, the series is convergent; if x > 1, divergent. If a; = l, the series==— 5 + 5— r + ^-sH- •■■ , and is convergent; see Ex. 2, 1>A u>4 0>O 104 CONVEEGENCY AND DIVEEGENCY. [CHAP. 6. — ^=1— -^T ir = ~.-, TT^; thus iim. —2-=0, and the series 13 M„-i \n |n-l n {n-iy ■«„_! convergent. 7. Here «„= . / -, and is ultimately eciual to unity; hence the series is divergent. [Art. 282.] M. (2«-l)a;"-i 2m- 1 _. -j. i 4.1, • ■ 8. — ^=7;; 7^ „ = 7i ^.ar. Hence if a;l, divergent. If x=l, the series=l-)-3 + 5 + 7 + 9+ , and is divergent. M„ (n + 1) n re + l/«-l\P j j.r. • i^.- * , , 9. — 2_ = i '. .; — / — and thus is ultimately equal «„_! ii" (n-l)P m \n + ly to unity. But «„= — — = — ^^ ultimately, hence we take for the auxiliary series the series whose n* term is — ^:^ , and this series is divergent except when p-l>l. [Art. 290.] Hence the given series is divergent except when ^i > 2. in «» *"'" "'"''' (™-2)2+l 10. — ^ = 7 nrrr -^ 1 stttt = , ,(2 , , ^ = a^. ultimately. Hence M„_i (»-l)^ + l (n-2)' + l {re-l)2 + l if a; < 1, the series is convergent; if a; > 1, divergent. T, , .,. • 1 1 1 1 1 If x = l, the series =1 + 5 + ^ + j7i+...+ —, — j+ ... Now -r, — r = -7; ultimately : hut the series of which this is the general n^ + 1 n' term is convergent [Art. 290]; hence the given series is convergent when x=l. 11. — — = -s-ri-, n2 , a=x, ultimately. Hence if x < 1, the series is convergent ; if so 1, divergent. m^-1 If a;=l, then «„= ^ .. =1, ultimately; hence the series is divergent. [Art. 282.] 12. — 2_ =3;, ultimately. And when x = l, m„ = 1, ultimately; hence the «n-l results are the same as m Ex. 11. 13- «» = ^^ = p^ = i • i • ""i^^t^ly- But the series whose general term is — is divergent exo^t when p>l; hence the given series is divergent except when y > 1. XXI.] CONVEEGENCT AND DIVERGENCY. 105 «» 14. — ^^=^1 ultimately. Hence if a; 1, divergent. If x = l, M„= — J— = — , ultimately; and the series ■whose general term is -J is convergent [Art. 290] ; hence the given series is convergent. 15. u„ = K^^)" - "^l- "= (« - 1)~". [See Chap. XX. Ex. 16.] Hence — — = 7-^^ — =^. ,- ,, = — - , and since this is less than X the series "n-i (e-l)-'"-" «-l is convergent. 16. Here Vn=- ;r — = -( I =-(l--) • " n" n \ n J n \ nj Thus '«n=- • «~^= - . ultimately. [Art. 220, Cor.] Hence the series is divergent. [Art. 290. Case 11.] 17. (l)«„=(«2 + l)*-re=n[l+^^ -n Hence the series is divergent. (2) Jn*+1- Vra* - 1 = , , = -ir-r-i = -5 . nltunately . Hence the series is convergent. 18 (1) Here u„= r = -. ultimately; hence the series is diver- gent. (2) The series=i + (^^ + ^) + (^^ + ^) ; the general term beins = 1 = -5 s- Thus the general term = — j- ulti- x-n x + n x^-n^ "" mately, and the series is convergent. 1 q 3t. = ^A^-'^y = 1 f^i-V = - , ultimately, whatever be the "• «„_j In ■ |to-1 n\n-lj n value of p: and this being less than 1, the series is convergent. 20. Let us compare the given series with the infinite geometrical pro- gression l + r+r'+r*+...; then if ^ be finite, the two series will be both convergent, or both divergent. [Ait. 288.] 106 CONVEEGENCT AND DIVERGENCY. [CHAP. Let -£=fc, so that «„=itr"; then !l/u„=r ^k=rk'^, so that li/u^=r, ultimately. Also if rl, ths auxiliary series is divergent ; hence the proposition follows. 21. The product, P suppose, consists of 2ra-l factors, and may be written P^tijU^u^ ... m„_i x =• , , 2n-2 2n-2 ^^^'^ "-'=2;rr3-2i:ri- Proceeding as in Art. 296 we have, log«„_,=log {l + (2„-3H2.-l) } =l°g (l+i) =i ■ "Iti^-tely; hence log P is ec[ual to the sum of a convergent series, and therefore P is finite. 22. When x=l, the general term T^+i in the expansion of n , „w J. n(n-l)(n-2)...(n-r + l) ' +'"' '^ 1.2. 3. ..7- • Thus the numerical value of T^j is .-. log T^, = Slog (l-'^), where S denotes the sum for all values of q &om 1 to r. When r is finite, T^+i is also finite ; when r is infinite, log 7^+1 is the sum of an infinite series, which is convergent or divergent according as the n + X infinite series - S is convergent or divergent. [Art. 296.] But this latter series is divergent ; hence we may write If n + 1 is positive, log T^i= - oo , and 2'^i=0; that is, the terms in the expansion ultimately vanish. If n + 1 is negative, log T^^ = + oo , and T^j = oo ; that is, the terms in the expansion become indefinitely great when n is negative and numerically greater than unity. IfTO + l = 0, 7i=-l,and(l + x)»=(l + l)-i = l_i + i_i + i_i + ...,^hieh is an oscillating series. XXI.] CONVERGENCY AND DIVERGENCY. 1U7 EXAMPLES. XXI. b. Page 252. 1 Here u - 1 • 3- 5 - (^"-7) ^J^ . 1. uere ""-2 . 4 . 6 ... (4»-6) ' 4n -4' «„ (4»-2)4« 1 1 ,,. , , • • u^, = (4«-5)(4»-3) • i-^ = x^ ■ ""lately- Hence if x <; 1, the series is convergent ; if x > 1, divergent. If x = l, then n{ — - — 1 ) = ,^ ' ^, ,. — , the limit of which is - ; hence the series is convergent. 2 Here 3.6.9... (B.^Tl) i. Mere ""-T.IO. 13...(3n + l) '^ ' M„ 371 + 4 1 1 ,,. , , .•. — — = — - — . - = - , ultimately. Hence if x<:l, the series is convergent; if x>l, divergent. If x= 1, n ( — ~ - 1 ) = -- = ~ , and the series is convergent. \«u+i / 3n 3 „ „ 22. 42. 62. ..(271- 2)2 „„ 3- Here ""= 3 . 4 .5 ... (21-1)27. '^^"' M. (2» + l) (271 + 2) 1 1 ,^. , , ••■ ^, = (27:)2 -^-2 = ^2. ultimately. Hence if x < 1, the series is convergent ; if x > 1, divergent. If X = 1, n I -^ - 1 ) = " ,„ „ = n , ultimately, and the series is con- \««+i / {2«)' 2 vergent. 7l»-i x»-i 4. Here «„=— j^; • _3L = I . _z . ± = . _ = , . - = — , ultimately. " «7.+i 1^ (tj + I)" a; (7i + l)"-i X /j IX""^ i" ex Hence if x<-, the series is convergent; if x>-, divergent. 1 «« « If a; = -. -7^ = - Tv^' (-0 - , ultim Hence the series is convergent. [Art. 302.] log 3l = I , ultimately. [Chap. XX. Ex. 17.] "n+i 2 108 CONVEEGENCY AND DIVEEGENCl' . [CHAP. [71 — 1 5. Here w,^= l .r"-i; .*. — - ■h= • „ , ■ - = — ^ •-=! + - ■- = -> ultimately. Hence if a; < c, the series is convergent ; it x>e, divergent. IN" 1 « / IN" 1 «n+i \ »/ « .•.nlog^-^^=«j»log(l + l)-li=n|«(l-i^,+ ...)-l|, the limit of -which is - ^ ; hence the series is divergent. p. 32. 52... (2)1-1)2 „ (2)1 + 2)2 1 1 _ ^, 6. "n= oa ^g -5 2 /o x3 ^ ; ■"■ = o , ,( 2 • " = - ■ ^iltimatelv. n 22.42. 62... (2n)2 M,^j (2« + l)2 a; a; ■' Hence if a;l, divergent. If x=l, ni — — - 1 ) = -K 7T^ . tlie Hmit of which is 1 ; we therefore pass to the next test. S ( '^n i\ Ji (-m-l)logre mlogn logjj in — ^ - 1 I - 1 [log )i = ^^ — r^ r-^ = J-,- = — ^ =0, ultimately. ( V«n+i / 1 (27! + l)- 4n2 4rj ^ Hence the series is divergent. [Art. 306.] _ (w-2 + a) (n-3 + a) ... (l + a) a(l-g) ... (7z-2-a)(ra-l-a) '• ""~ 12. 22. 32... («- 1)2 ' .■. — ^ = ; i T-, r=l, ultimately. «»+i (™-l + a){»-o) )t (m - a + a2) ^«n+l / = 1, ultimately. (n - 1 + a) ()i - a) ()i - are + a2re - o + a^) log n I V«n+i IS (»-l + a)(re-a) _ (1 - a + a2) log n Hence the series is divergent. 8. Here «"=^^re— = «„ _ 1"-''-^ (a + 7?a)" "■"n+i" l!* (a + 7i + l.x)"+i = 0, ultimately. (n + l)?!".!;" 1 1 ,,. , , I 1 + - 1 X XXI.J CONVEEGENCT AND DIVERGENCY. 109 Hence if x <: - , the series is convergent ; if a; > - , divergent. This result it will be observed is quite independent of a, and if we put a = 0, we obtain x + -;;- H — '—- + ..., which is the series discussed in Art. 302 ; hence the conclusions obtained in that article hold for aU values of a ; thus when x=-, the series is divergent. n „ ^ tt(a + l)...(a + »-2);3(|8 + l)...(j3 + n-2 ). " 1.2.3... (n- 1)7 (7 + 1) ...(7 + 71- 2) ' .-. — — = ; ' , , , — - — ;- =1, ultimately. «n+i (a + n~l){^ + n-l) f ■"„, ^\ H=(7-a-^ + l)-n(a-l)(/3-l) „ , ,^. ^, n —^-1 ) = —^ — - — " — 4vi — h; — iir^- ^=7-a-8 + l, ultimately. V^n+i J (« + a-l){n + ;8-l) T P , J Hence if 7 - a - /3 is positive, the series is convergent ; if negative, divergent. If 7-a-|3=0, thennfi^-l") = ,"'"" ^''"/"^~V. = 1. ultimately. V«„+i J (n + a-l)(n + si-l) ^ („ f Jf» _ l\ _ 1} lo» „= - {»( ''-l)(/3-l)+"(«+<3-2)-(a-l)(^-l)}log,t ( V«»+i / I ° (n + a-l)(n+p-l) Hence the series is divergent. 10. .„=x-{log(«-M)}'= .^„^ = gg±f .i^i.ultimately. Hence it x < 1, the series is convergent ; it j > 1, divergent. ""-'■«„+i-llog(n + 2)[ - i„g„+?_ . 1+2 (,71 ) \ 7llOg7lJ = ('1 -L_Y=l__2 ; .-. „('ii''--l')=--?-=0, Ultimately; \ nlognj (ilogji Vh„4.i / log k hence the series is divergent whatever be the value of q. 11 Here a(<^ + l)(^ + 2) (a + .-2). " 71-1 i = 1, ultimately. '""n+l |»-1 "a + 7t-l 71 + 0-1 „ C JL» _ 1 ") = ~"^''~-^) = - a + 1, ultimately. \"n+i / " + a-l Hence it a be positive, the series is divergent; if negative, convergent. If a is zero, the series reduces to its first term 1 and is convergent. 110 UNDETEEMINED COEFFICIENTS. [CHAP. 12. -^=1, ultimately; n — ^ - 1 = r-^ — TT — ^r-T-a =A-a, ultimately. Hence if ^ - a > 1, the series is convergent ; if ^ - a < 1 , divergent. If^-a=l,then^f^-lV'^ + p-'']r+-; = (B-b-a) -^^ = 0, ultimately. Hence the series is divergent. It should be noticed that the result is independent oiB, b, C, c, .... EXAMPLES. XXII. a. Page 256. 1. Let 12 + 32 + 52+. .. + (2«-l)2=4 + Bn + Cn2 + Dn3 + ... then 12 + 32+52+.. . + (2»- 1)2+ (2)^ + 1)2 =4 + B(ra + l) + C(« + l)2 + D(„ + l)3+... .-. by subtraction, (2n + l)2=B + C(2« + l)+D (3n2 + 3m + i) + ... .-. the coefficients after D all vanish, and on equating coefficients of like powers of TC, we have B + G + D = l, 2C + 3I> = 4, 3D = 4; .-. Z) = |, C=0, B=-\; .: S=A-\n + ^nK Put»i=l; thus we find ^ = 0; hence S=-^{in^-l). 2. Let 1.2.3 + 2.3.4 + 3.4. 5 + .. .+m(re + l)(» + 2) Then as in the last Example, we find (B + l)(n + 2)(n + 3)=B + C(2» + l) + D(Bji2 + B„ + i) + j3(4„3 + 6n2+4„ + i). 1 3 11 S Equating coefficients, we find E = -, D=-, G = — , B = -- 4 2 4 2 SJ . " 11 n 3 ., 1 . = 4 + -» + -^»2+ „3^ „4. J 4 2 4 When m = l, ^ = 0, and S reduces to " (»+ 1) (" + 2) (» + 3) 4 XXII.] UNDETERMINED COEFFICIENTS. Ill 3. Let 1.2'' + 2.d^ + 3.43+...+n{n + l)^=A + Bn + Cn'' + D'n?+En*; then as before {n+l){n + 2)^=B + C(2n + l) + D{Sn' + 3n + l) + E{in^ + en'> + 4:n+l). Equating coefficients, we find B=i D = l, C=-, B = %; 4 6 4 6 o 4 6 4 When n = l, ^ = 0, and S rednces to ri{n + V)(n + 2)(Zn + 5) _ 4. Let 13 + 35 + 53+. .. + (2»-l)3=4 + Bn + C»2 + Dm3+£«4;tlienwefind (2ra+l)5=B + (2n + l)C + i)(3n2 + 3n + l) + E(4ra3 + 6n=' + 4n+l). Equating coefficients, we find £ = 2, D = 0, C= -1, B = 0; Whenm=l, ^ = 0; hence S=?i2(2„2_i). 5. Let l* + 2« + 3*+...+n*=4 + Bn+C«2 + Dn3 + En^. Then in the usual way we get («+l)-»=B + (2n + l)C+(3;i2 + 3re + l)D + (4?i3 + 6m2+4n + l)E + F (5n* + lOn^ + 10n'> + 5re + 1) ; whence by equating coefficients we obtain ■■• ^=^-30"+3" +2" +3"- When »= 1, A = 0, and S reduces to ^ (k + 1) (2n + 1) (Sn^ + 3re - 1). 6. Assume x^ - 3px + 2q = (x + k) {x' + 2ax + a^) . Multiply out, and equate coefficients of like powers of x ; thus k + 2a = 0, a^k = 2q, -Sp = 2ak + a". Eliminating k, we have q= ~a^, p = a^; whence it follows that p^ = q^, which is the required condition. 7. Assume ax^ + l>x''+cx + d = (px + q)^. Equate coefficients of like Powersoft, and we obtain p'= a, Sp^q = b, Spq'=c, q^=d, whence V=27aH, and c'=27ad''. 8. Assume a''x* + hafl + cx^ + dx+f'=(ax'+px+f)^; whence, by equating coefficients of like powers of x, we obtain b=2ap, c=p'' + 2af, d=2pf; :. ad = hf, and "=(2^ +2o/. 112 UNDETERMINED COEFFICIENTS. [CHAP. 9. Assume ax^ + 2'bxy + cy^ + 2dx+2ey+f={Ax + By + C)^. Then A^=a, B^=c, (?=f, AB = b, AG=d, BC=e; whence the required conditions foUow at once. 10. Assume ax" + bx^ + cx + d=(x^ + 'h?) iax + j^\ ; equate coefficients of like powers of x, and we obtain & = ^ , c=ali^; .:- = -, 01 tc = ad. 11. Assume (i?-5qx+ir={x^-2xc + d') (a? + cu?+bx+-^\ . Multiply out, and equate coefficients of like powers of x ; then we have a-2c = 0; c''+b-2ac = 0; ac''-2bc + -^=0, bc'=5q. From the two first of these, 30^=6; substitute for 6 in the remaining equations, and we easily find r=c'', j=c^; .•. r*=q^. 12. (1) is 8'° equation of the second degree satisfied by the three values a, b, c, as we easily find on trial. Therefore the equation is an identity. (2) is solved in the same way. 13. If ax'' + 2hxy + by' + 2gx + 2fy + e={px + qy+r){p'x + q'y+r'). We have, by equating coefficients, pp' = a, qq' = b, rr'=c, qr' + q'r=2f, rp' + r'p = 2g, pq'+p'q = 2h. Multiply the last three results together ; thus 2pp'qqW +pp' (i^r'^ + g'V) + qq' (pV^ +p'H^) + rr' {pY" +py^ = Sfgh. 2abc + a{ip- 26c) + b {ig' -2ca)+c{ih'- 2ab) = 8/<7?i, which reduces to abc + 2fgh - af^ -bg^-ch^ = 0. 14. We have ^=lx + my+nz, also a; = Zf +mi; + nf, y=n^ + lri + m^, z=m^+nri + l^; .: , by substitution, we have the identity J=Z(J| + mi; + nf)+ro(mf + J») + mf) + n(mf + B?; + Zf). Whence, by equating the coefficients of |, ij, f on the two sides, we obtain the required relations. 15. The sum of the products is the coefficient of a;"" in the expansion of {x + a){x + a^) [x + a?) ...{x + a^). Let (x + a)(x + a^)...(x + a'') = x^ + AjX'^-^+... + A„.^^-i^x'^^ + A^_rX'<-+.... 1 Write - for x, then since - + a'"=- (as + a'^'), we have a a a ' i (x+a^) ^x^a^) ... (x + a--^^) = (^lJ+A, g)""' + ^, ©""' + - ' XXII. J UNDETERMINED COEFFICIENTS. 113 .: (x + a){ai«+ A^ax"^^ +...+ A„_^_^ a""*^! x^^ + ^„_^ o""'' x' + . . . } = (x + a''+i){a;» + ^ia;''-i + ...+^„_,.ia;'^i + .4„_,x'-+...}. Equate coefficients of a;'+^ ; then A^_^ a"-'- + J„_^i 0"-'= 4„_,.i o-^i + 4„_^ ; .-. A„., (a"-'- - 1) = ^„_^_i a'^-r (ar+i - 1) ; that is, put r+ 1 for r, then A^ = A,a a''-i-l jii = a — —= , since i4|, = l. Now multiply these resxilts together and cancel like factors, and we easily obtain A^_^ in the required form. EXAMPLES. XXII. b. Page 260, 1, Let z ^=af) + aiX + a0' + ai3ifi+ ...; then 1 + 2a; = (1 - a; - a;'^) (a^ + a^x + a^ + a^ + ...). Then 1 = 05, 2=01-0,,; whence aj = 3. The coefBoients of higher powers of X are found in succession from the relation a„ - a„_i - a„_2 = ; hence u„=4, and 03 = 7; thus , "^ '^ „ =l + 3x + 4a:° + 7a:^+... 2, With the same notation we have 1 - 8a! = (1 - X - 6a!') (aj + a^ + Oj^'^ + Sga!' + ...). Then a^=\., ai-Oo=-8; whence ai=-7. The other ooefBcients are determined in succession from the relation a„ - o„_i - 6a„_2= 0. 3, Wehave l + a!=(2+a! + a;2) (Oo + aia; + a2a;2 + ...). Then 00=5, 2ai + a|)=l; whence 01=2-. Also for values of n > 1, 2a„ + a„_i + a„_2 = ; 11 3,-31 1 •■• 2a2=-2-5>o^»2=-8! ^'^^ 2a3 = g-j; or ai = z^\ 1+x 11 3,1, ■■ 2 + x+a:'' 2 4 8 16 Example 4 may be solved in a similar way. H. A. K. 114 UNDETERMINED COEFFICIENTS. [CHAP. 5 . Let the required expansion be 65 + Sja; + h^^ + b^ +... Then l=(l + ax-ax^-x^){\ + bj^x + b2x''+bsX^+...); .: b„=l, 6i + 6„a=0; whence 61= -a. Also b^+bja-b„a = 0; whence 63= «(»+!)• And 63 + 620-610-60 = 0; whence 63=1- 2a2-a3; 1 + ax-ax^-x' \ I \ ' I , . 6. By putting »i=l, 2, 3, ... successively we see that the required expan- sion will have ooefSoients 1,4, 7, 10, . . . ; that is, a + 6x=(l-x)3(l + 4a; + 7rj;2 + 10a;' + ..:); .■. by equating coefficients we have o = l, 6 = 4-2 = 2. 7. As in Example 6 we find that a + bx + cx^={l-xf{l + 2x + 5x^ + 103i?+...); .: a=l, 6= -3 + 2, c = 3-6 + 5; or.a = l, 6=-l, c = 2. 8. Since y = Q when a;=0, we may assume ■y=AjX + A2x''+A^x^+...; substitute this value for y in the given relation ; thus (AiX+Aj,x''+A^''+ ...)^ + 2{AjX+A^^ + AsX^+ ...)=x{l + AiX + A.^'^+ ...). Since this is an identity, we may equate the coefficients of powers of x; thus we obtain 2.4j = l, or ^1=5; ^1^ + 2^2=^1, whence ^2=i; ^ 8 2AjA2 + 2A3=A^, whence ^3=0; AJ' + 2A:iAg + 2A^=A,, whence ^4= - -— ; 1 1,1.. •■■^ = 2^ + 8^-128^+- 9. Here y=0, when x=0. Also y changes sign with x; therefore we may assume ^=-^iy + -^sy^ + -^By^ + Ajy'' + ... Now proceed as in the last Example, and we get c(Aiy+A^'+A^-^ + ...)'> + a{Ajy + A0^+...)-y = O; .: equating coefficients, dill- 1 = 0; or^i=-; cA3 + a.43=0; whence .4,= -.^. a^5 +30^1=^3 = 0; whence ^= ?^. 4 = —. a° a^ a' aA, + ScA-^^A^ + ScA^A^^ = ; A = -— - —-— I £!. - ^2c3 '^ a'a^'a' a 'a' a? a" ' and so on; thus :.=^ _ ?^% ?^ _ 1!£Y XXII. J XJlirDETEEMINED COEFFICIENTS. 115 113 Now put c = l, y = l, a=100; then a!=j^ - — — j + Tj^,- ... becomes tlie solution of a;'+100x-l = 0; .-. x = -01 -•00000001 + ... = -00999999 approximately; also since the first 3 term rejected is ^ , and this when expressed as a decimal begins with 13 ciphers, the value found for x is accurate to 13 places of decimals. 10. Assume {1 + x) (l + ax) {1 + a^x) ... = l + A-^a; + A2X^ +... Change x into ax ; then we have, as in XXII. a. 15, (l + x)(l + A^ax + A^a^x^+...) = l + AjX+A2X^+... Equate coefficients of x"* ; thus ^, o*" + 4y_i a''-i = 4, ; .-. ^,(l-a'-)=^,_ia'-i; •'■ "^ ~ 1 - a"- '•-1 - (1 - a"-) (1 - a>-i) ^"-^ ' (l-a-Kl-a'-ija-g^-^) ^'•-2= ^""^ '° °°- (jl+2+3H-...+(i.-Il ^^"^ -^'•= (l-a^)(l-a^-i)...(l-a) ^» , since Ad = 1. {l-a){l-a?)...{l-ar)' 11. Let the expansion be A^+A-^x + A2X^+...+A^x''+.... Multiply each side by 1 - ax ; thus (l-a''x)'l-«^x)... =^'' + '-^^-^°°^'^ + -+^^"-^'-''')^"+- But by writing ax for x we see that the expression on the left =A^ + Ajax+A^aV+,.. + A^a'^x^ + ...; aA ■■■ ^»o"=K-.4„_ia); thus A^=^-f^,&o. And finally A^= (l-a) (l-a=) ... (l-a") " 12. (1) Proceed as in Ex. 2, Art. 314, and we find - — -. ^ — + \. ' .—. T ... to re terms In + l Ji+1 2 \n + l =the coefficient of x"*^ in(x + — +|3 + ..,l =^n. 8—2 116 UNDETERMINED COEFFICIENTS. [CHAPS. (/p2 ^3 \ n-f-1 K + T^ + v^ + . . . I = »"+! + terms containing higher powers of x. Expand the left-hand side and multiply all through by e-*; then eiw_(„+ l)e(n-i)":+(±tll^ecn-2)a!_... to M-l-2 terms = e"'" (s^+i-l- ...). 1.2 The last two terms of the series on the left are ( - 1)" (« + 1) c» + ( - l)''+i c-'^ ; .•. the coefficient of a;" in the series is 'I" / n (»-!)" (n + l)n (71-2)" ^ ^ , ,,„., (-1)" i--(m-l-l)5— i — ^ + ' ' . ' , ' -■■■ toTO terms + (-l"+i^-r-^ ; Iw'Ire 1.2 k ^'|k' and this is equal to zero, since on the right-hand side there is no term con- taining a:". Transpose and multiply up by \n. (3) We have e"^ (l-c^)»=e^-7?e2"i-)-'L^L:ili gsa; _ _ t(J n+1 terms; .•. the coefficient of x" in the expression on the right is _1_ 2^ n{n-l) 3« [re~"'|ra"'' 1.2 ""jm' Again the expression on the left is ( - 1)" e* (e" - 1)", which may be written (-l)"•(l + a:-t-...)(a^^-...); thus the coefficient of a;" is (-1)". Equate the two coefficients, multiply up by \n, and the required result follows. (4) eJ'^(e'"-l)'» = c»"^ |eM:_„e(n-l)a;_|.!L^i:il)g(n-2)a:_ _ I 1 . ^ Equate the coefficients of a;", and the required result follows. EXAMPLES. XXIII. Pages 265, 266. 1. Assume:;- f~,fto = ,—?:-+ ^ 1 - 6a! -1- 6a;2 - 1 - 3a; ^ 1 - 2x ■ Then 7a;-l = ^(l-2s)-fB(l-3a;); .'. equating coefficients, A + B= -1, 2A + 3B= -7; whence A =4, B=-5; ■'■ l-5a;-l-6x2~l-3a; 3T2S' „ . 46-hl3x A b' 2- Assume ^^^, _ n^ _ 15 = 4^^ + g^Ts ' Then 46 + 13a;=4 (3a;-5)H-B(4a;-l-3); XXII, XXIII.J PAETIAL FRACTIONS. 117 .-.34 + 45 = 13, -54 + 35=46; whence 4= -5, 5 = 7; . 46 + 13a: _ 7 5 ■ ■ 12a;-''-lla;-15 ~ dx^ ~ ix + i ' -i g--- l + 3a; + 2a:2 _ (l + 2a;)(l + 3;) _ l + 2x {l-2x)(l-x') (l-ix)(l-i?)- (l-2x)(\-x)■ ^ \ + 2x A B Assume r- — „ , „ — , = 1 • (\-'ix){l-x) l-ix^l-x' .: l + 2a;=4 (l-a;) + B(l-2n;). Puta!=l; then 3= -B; also4+B = l; thus 4 = 4. ic^ — 10a:+13 ABC *■ ^'''"^' (x-l){x-2){x-3) = ^31 + ^32 + ^^3 ; ^l^^'^"'^ ^y Plt«"g x=l, x=2, ir=3 successively, we get 4 = 2, B = 3, C?= -4; ,, .234 .'. the expression = r H . ^ a;-l a;-2 x-3 5. Dividing out we have -; ^ytjt ^=1 + x(x-l){2x + 3) a; (a; -1) (2a; + 8)" 2a;-3 A B G ^°^ ^^^^^ :.(x-l)(2x + 3) = ^+^3l + 2^T3' ^^ ^"^ 4 = 1, £ = -1, C=-|; ti, -1,1 1 8 .-. the expression = lH ^, =^ - j-tjt -r . a; 5 (a; - 1) 5 (2a; + 3) 7a; — 4 7. By division we find the given expression = a; - 2 - 1^-^ jr . (a; + l) \x — Of Ix-i A- B ^ C . , Assume ■; -r-r-, T- = =■ + 7 rr; H ; wB hnd (a; + l)^(a;-3) x + 1 (x + l)^ x-6' 17 11 17 ^--I6'^-T' '^-le- •■• *^^ «^P™^"°°="'-2- 16(^) - 4(F^+ leFTi)- 26a; (a; + 8) ^^ 8. Theexpression=p-p^!^^-^^. Now assume (-S»wJT5)=^^?Jf'*^-^ + «=^(^^ + ^) + (^^ + ^)(^ + ^) = . 3 Tj 3 ^ 41 whence 4 = 2g, B=-^, C-^. 26a;^ + 208a; / 3 3a;-41 \ ' (x^ + l)(a; + 5)~^V^ + 5 x^+lj' 118 PARTIAL FRACTIONS. [CHAP. „ , 2a;2-lla; + 5 A Bx + G 9. Assume , px,„a , „, ., = ::^ii + : {x-S){x^+2x-5) x-3 x^+2x~5 Then 2x^ -llx+5={A+B)x^ + (2A -3B + G)x-(5A + 3C) ; whence by equating coefficients A= -1, B = 3, 0=0 ; ., . Sx 1 .-. the expression = -= — tt = 5 ; 3c' + 2x-5 x-6 10. Put a; -1=2 ; then the expression ~ 2* ~ z* ~ z z^ z^ z* 3 17 5 x-1 {x-lf {x-lf (x-1)*' 11, Put the expression = r + ^ ^ . ^ , and proceed as in Art. 317. "We thus find A = l, and / (x) = 1 - a'. 23 3 1 113 3 2 .-. the expression = =■ + — ^ - — + - x-1 x + 1 {x+lf (x + \f^ (x + lf 4 1 12. The expre3sion= . - , and the general term is ^^'{4.7'--4'-}a;'-. 11 4 13. The expression = .^_ . - „ ,. . , and the general term is iki-izm 3 X 2>-i I a;'. !*■ x2 + 7» + 10=^-x^ + 7x + 10=^+3^JT5)-3F^- ^he general term is ( - l)' . g -jg^i - ^^j}- x'. 114 15. The expression= - =—— +— , and the general term is {1 _ ( _ l)r _ 2'^-2} a;"- or {1 + ( - l)'^i - 2''+2} cc''. XXIII.] PARTIAL FRACTIONS. 119 16. The e^pres3ion=3-^jl2^-g^^ + ^-^, and the general term is - {9r + 8 + ( - 1)'-2''+2} a;'". 17. The expression= J— -— + ^— — — , and the general term is 4'-1(12 + 11j-)!c''. O O (3 18. The expression = — — + ,., , ,„ - „ , „ . The general term is 1 + x {1 + xy 2 + 3x (-ir|3r+5-^f ^'•, 3 1 — 3a; 19. The expression = ^ ^_^ + ^.j^^^^) ■ The general term is 2{(-l)a-3}a;'-if riseveu, and - 1 {1 + ( - 1)"^} a;'- if r is odd. on Tt 1 i-u ■ z + 2(l-z)2 2-32 + 222 2 3 2 20. it 2 = 1 - a;, the expression = —. = = = — , — , + - z^ z^ z' z^ z = 2(1-k)-3-3(1-!c)-2+2(1-x)-1; .-. the coefficient of k'" is (r + 1) (?• + 2) - 3 ( r + 1) + 2, or r^ + 1 . «., . 1 ABC 21. Assume j- -j- ,.,.,. : = :; + 1j — :r- + = ; (l-aa;)(l — oa;)(l-ca:) 1-ax 1-bx 1-ca; .-. 1=4 (l-bx){l-cx)+B(l-ax){l-ex) + C(l-ax){l-bx). By putting in succession 1- ax = 0, l-bx = 0, 1 - cx = 0, we find that A = -. jT-T T . and similar expressions for B and C; {a-b){a-c)' .: the required term is the (r + l)"' term in the expansion of -(l-ax)-'-+ + {a-b]{a-c)^ ~[(a-b){a-c)'^ {b-cj(b-a)'^ (o-a)(c-b)\^'^' V 3-2x^ _ A B O D ^^- ■^"'i (2-3x+a;y~(2-i!:)2"''2-x^(l-9;)2'''l-a:' .-. S-2x'=A{l-x)^+B{2-x){l-x)^+G{2-x)^+D(2-xy(l-x). Then 4 + 2B + 4a+4D = 3; -B-D=0. Also a:=2 gives 4= -5; and a; = l gives 0=1. .: 2B + 4D = 4, and B=-D, whence D=2, B=-2; 120 PARTIAL FEACTIONS. [CHAPS. 3-2^2 5 1 1 2 ¥?) 'i ,^+M-.r^2'*"l ••■ (2-3x+a:2)2- / xV x^ (l-x)"^ l-x' .-. the coefficient oix'^--/-^- - +(r + l) + 2 = 3 + r-- -^. 23. (1) The re"" term is -p, ^^r-yr ^^ , and this may be put in the (J. -J- X J ^J. + X j ''°™ X (1 - x) {l + »"+! ~ r+^j • Similarly each term of the series may be decomposed, and on addition we find that all the terms disappear except one at the beginning and one at the end ; thus . 1 f^^ Li x(l-x) (l + x''+i l + xf ■ a''-ix(l-a"x) (2) The n* term is (1 + a^-ix) (1 + a»x) (1 + a^+ix) ABC Assume this= + ., , „ + , , „ , , ; then in the usual way we find 4 = - -5 = C, B= , ^,.. . Thus the rf" term (a -1)2 (a-iP ^ 1 ( 1 2 1 1 ~(a-l)2| l + a''-ia;"''l + a»x l + a-'+ixj * If we decompose each term of the series in this way, we find on addition that all the terms disappear except two at the beginning and two at the end. Thus the SUm = -; — -,-,-,■ \- :; — r; ^; h = V . (a-lp (l + a"x l + a"+ix 1 + x 1 + axJ 7.2n-2 1 f 1 1 1 24 The»"»term = = = ) - - ( (1 - x2"-i)(l - a^-i+i) a; (1 - x^) \l - a;2"-i 1 - a?»+ij ' Thus the series may be written x(l-x2)|l-a; l-x3"'"l-x3 1 _a;6 + - *° ^^j 5 and this reduces to — ^= — r-. X (I - X) (1 - x^) 25. The »"' term can be put in the form Jl r x» 2x''+i x't+s -j (i - xf [i^^ ~ i-xo+i "*" 1 - x"+-J ' and, as in Ex. 23, the sum is found to be 1 ) X _ x2 x'l+l x''+2 I XXIII, XXIV.] RECURRING SERIES. 121 26. We have p. r^- — j-r— :-=l + S-^x + S^^+ + S„a;''+ where S„ = the sum of the homogeneous products of n dimensions which can be formed of a, b, c and their powers [Art. 190]. Assume j- rj^ — =-^— r = = + ,= — =— + = ; then by putting (l-ax){l-bxj{l-cx) 1-ax 1-bx 1-cx 0^ l-ax,l-bx, 1-cx equal to zero successively, we find A = , r~, : , and {a-o)(a — c) similar values for B and 0. -(l-aa;)-i + + ...; " (1 - ax] (1 - bx) (1 - ex) (a -b}(a~ c) ^n+2 J^n+'i c^+^ .: the coefficient of x" is -, =^7 \ + Ti rn r + 7 r? — m > 'which {a-b){a-c) (6-c)(6-a) (c-a)(c-6) may easily be thrown into the required form. EXAMPLES. XXIV. Page 272. 1. Let 1-px-qx^ be the scale of relation; then 13 - 9p + S^ = 0, 9-5p4-3 = 0; whence ^ = 2, q = l. Nowlet S = l + 5x+ 9a;2 + 13a;3 + ..., then -2xS= -2x-10x'-18x'- ..., x^S= x^+ 5x3+...; .-. S(l-2x + x^) = l + 3x; ■••^=i4S^=(i+^-)(i-)-' = (l + 3a;){l + 2a; + 3r>;2+... + (r + !);!;'•+...}; .-. the general term=(3r+r + l)a;''=(4r+l)ic''. o TT 1 o o 2 + a; 1 1 2. Herei,= -1, ) The sum of n terms = ' _ — i_2x ' 8. The scale of relation is 1 - 7a; + 12a;2. The generating function 2- 7a; 1 1 l-7a; + 12a;2 l-4a; l-3a; The «"" term = (4"-i + S""') a;"-! ; .-. the sum of n terms = S4''-ia;'^i + Z3"-ia;''-i=^--j^+ ^^^^ . 9. The scale of relation is 1 - 6ii; + lla;^ - 6a^. The generating function l_4a; + 5a;2 111 ~'{l-x)(l-2a;)(l-3a;) 1-a; l-2a; l-3x' The m"" term = (1 - 2^' + 3"-^) x'^^ ; , , 1-ai^ l-2"a;'' l-3»a;'> XXIV.J EECUEEING SERIES. 123 3 10. The scale of relation of the series - ^ + 2x +0x^ + So? + ... is 1 - 3a; - 4a;2, and the generating function is 18a; -3 or iJj^ m. L0ll-4a; 1 + xf ' 2(1— 3a;-4a;2)' 10 1 (8 O^'i-si .-. the «"> term = j^ {4"-! - ( - 1)"-! 16 } a;"-! = 1 ; ( - 1)" + g \ »""'• Put x = l, then the m"" term of the given series = - {8(- l)" + 22"-3} ; and the 5 4 1 sum to n terms = ^{(-1)"-!} + ^ {2^-1). 11. If we denote the series by ■U1 + U2+ ... +«„, we have in the first case %-«u-i=2n-l, ■(6„_j-«„_2 = 2m-3; .•.!*„- 2m„_i + M„_2 = 2 = M„_i - 2 ^^^ "M+i-f"ji=9''n-i> *his result agrees with that in Art. 325. 13. The scale of relation is 1 - 3a;2 + 2x3, and this = (1 + 2x) (1 - x)^ ; 3 — a: + 4x^ 2 12 .-. we find the generating function = ^ _ 3^, ^ ^^^ = ^^-^^ " TT^ + r^ ■ Hence the m"' term = { 2m - 1 + ( - 1)™-! 2"'} x". 124 RECURRING SERIES. [CHAPS. Put a;= 1 ; then the sum of m terms = (l + 3 + 5 + ... +2m-l) + {2-22 + 23-... + (-l)™-i2"} = 711^ + 2 .- 1 + 2 .-. the sum of 2«+l terms=(2n + l)2 + - (2»'+i + l). 14. Let the generating functions of the two series be - A B + px + qx'' and l + rx + sx^ respectively ; then the sum of the two infinite series is A -, + -. B 1 +px + qx^ 1 + ra; + sa;^ This is therefore the generating function of the series whose general term is (a„+6„)a;'', and on reduction we find the generating function has for its denominator l + (p + r)a; + {j + s+pr)a;^ + (g?'+2)s)a^ + 2sa;*, which is there- fore the scale of relation of the new series. 15, Let the given series be Md + «j + M2+ ..., and let the scale of relation contain & constants ; so that Let S„=tti+«2+M3+. ..+«„; then S„-S'„_i=M„ =Pl"»-l+2'2«»i-2+ — +i'ft"n-* ■■• S„= (;)i+ 1) S„_i - (ill -jps) S„_2- (P2 - Ps) "^n-s - •■• - (P*-l -i's) 'S'n-J: -PhSn-ic-l- Thus if M„ is formed from the preceding h terms of the series 11^ + 11^ + 11^+ ..., S„ is formed from the preceding k + 1 terms of the series Si, S^, S^ EXAMPLES. XXV. a. Pages 277, 278. Examples 4 — -11 niay be worked as in Art. 333. It will be sufficient here to give the two following solutions. 6. 3 1189 3927 3 109 360 10 33 8 1 8 .'. the successive quotients are 3, 3, 8, 3, 3, 3, 3; and 1189 _J_J^ Ji_J^A J_i. 3927~3T3+ 3+ 8+ 3+ 3+ 3' 1 3 10 33 and the first four convergents are - , Yo ' qq ' Tm ' XXIV, XXV.] CONTINUED FRACTIONS. 126 10. -3029 = 3029 10000 ■ 3 3029 10000 6 200 913 2 32 43 10 10 11 1 .-. the snooessive quotients are 3, 3, 3, G, 1, 2, 1, 10 ; and 11111111 •3029 = 3+ 3+ 3+ 6+ 1+ 2+ 1+ 10' 1 ^ in R^ and the first four convergents are - , — , ^-^, — . 12. A metre = 39-37079 x ^ yards =1-0936 yards. Also 1-0936 = 1 + _J^ 1_ J^ 1_ 1 10+ 1+ 2+ 6+ 6' 11 12 35 The convergents are 1, -^ , — , ^ , . . . . Thus 32 metres are nearly equal to 35 yards. 13. The continued fraction corresponding to -24226 is 11111 4+ 7+ 1+ 4+ 1 + and the fii-st five convergents ^^^^ j • 29 ' 33 • iei ' igi ' 14, The continued fraction corresponding to -62138 ia J_J:--1--L-LJ_JL J_ 1+ 1+ 1+ 1+ 1+ 3+ 1+ 2+ 1 2 3 5 18 23 64 and the convergents are 1, 2' 3' 5' 8' 29' 37' 103' 15. 162 parts of the first scale are equal to 209 parts of the second scale. 209 .-. 1 part of the first = ^jg parts of the second. Convert :rw^ into a continued fraction and we find 1 + 5 — = ; — -= ; and 10.3 4Q 0+^+4+5 the fourth convergent is ^ . 40 In other words, 1 part of the first scale is nearly equal to -^ parts of the second ; that is the 31" division of the first nearly coincides with the 40"' of the second. 126 16. CONTINUED FRACTIONS. [chap. n + 1 n^+n^-1 n^ + n' + n^ + n TO + 1 n+1 m + 1 -nf-n - 1 — n^-n'-n-l n- 1 n-1 Thu3 the continued fractionis™-l+ ■; tt — -; ^ -- (ra + l)+ (re-l)+ n+1 , n-1 m" n^-n^ + n-1 m'+m^-l vergents are . =-, = , — — — . n n + 1 n^ n^+n^ + n + 1 , and the oon- 17. (1) The expression on the left = fe?^2-t^2=^— ^2=1 = ??>?« =^_ (a>.?n + 2n-l) - 2n-l O-n In in A i'n Pn ' ■^ Pn-l^ "'n¥vPn+Pn^\-Pn-\ _ <>'r+-iPn P»+l PiU-l Pn+1 ' 18. We have i)„_ig„-2-p„-22n-i=(-l)"-^- [Art. 338.] Again, Pn^n-i - Pn-2in= {"'nPn-l+ Pn-a) 2»-2- (an?n-l + 2m-2)i'n-2 = an{Pn-l 2m-2 -i'B-22n-l) = (- 1)"~^0„. Similarly, i'^iSn-2-i',i-23n+i = «n+l (i'u3n-2 -iln-Z Sm) + (Pn-1 2n-2 -^-2 ?n-l) =(-ir-M«^i«„+i). Finally, ^',^^-2^n-2-i'«-2^»+2 = »^2 iPn+l ?n-2 - i'»-2 ?ji+l) + (P« ffn-2 - i'ii-2 Sn) = ( - 1)""^ («n+2 0«+l »» + an+2 + «n) • EXAMPLES. XXV. b. Pages 281, 282, 283. 11 12 35 222 1367 1. Theconyergentaarel.j^, H' 32' 203' 1250- C^^^- a- 12-] .". m 222 1 1 203 • *^' '"°' ^' " pip- ^^^ >27l250p • 2. The convergents are 1, 3, jg, — ; the fourth convergent - 1 , . , . 1 — ;^ , Which IS <7^7™ , or -0001. '115 differs from the true value by ■ (115)2 "(100)2 XXV.] 3, 1-41421=1 + CONTINUED FKACTIONS, 111111 127 2+ 2+ 2+ 2+ 2+ 2+ 3 7 17 41 99 239 ; the convergents are 1, 2' 5' 12' 29' 70' 169" 99 1 .•. the error in taking ^~ as an approximation <=^r — r-7--. 70 ^ 70 X 169 [Art. 340.] a+1 a + 3 a3 + 6a'-'+13a + 10 a? + 5a?+ 7a a^+ 6a + 10 a?+ 5o+ 7 a+ 3 a+ 3 a« + 6a3 + 14a2+15a+7 a« + 60S + 13^2 + 10a d'+ 5a + 7 a2+ 5a + 6 a + 2 .■. the fraction = 1 a + 1 a+ (a + l)+ (a + 2)+ a+3 a= + 3a + 3 5 , and the first 3 convergents a' a^+a + l' a3 + 3a2 + 4a + 2' 5. Pn F«-l_(-l)'' = -^, -?a_£_^=__, -™_^:2ti = }_i^; add these results together. 6, We have 2233 Pn _«nPn-l+Pn-2 2n 2n-l 2»2n-l i'u-2. i'n-1 Pn-1 = »« + Pm-I and ^^i=„„_, + ^P2=3; ^P2!^=a„_,+^^^;andfinally?? = a„+i. Pn-a i'n-2 Pn-3 A-S Pi ' ^l Thus Pn , 1 1 1 1 Pn-1 «n-l+ ««-2 + a2+ Oi SimUaily the second result follows, for —=03. 2i 7, (1) We have to prove that PnPr^2-Pn=^Pi,^l'-PM-lPn-l> °^ Pn{Pn+2-Pn)=Pni-l{P„A-l-Pn-^- Now Pn+2= SO thatp^i-^„_i=ap„, whence the required result easily foUows. (2) is the particular case of Ex. 8, when b=a. 8. By trial we find the required results hold in the case of the first few convergents. Assume that _ a ?2«-2 —Pin-l I ?2n-S — J P2n-2 > 128 CONTINUED FRACTIONS. [CHAP. then 22»= *?2n-l+ ?2n-a= * ("^3211-2 + ?2n-s) + ?2n-3 = a (bP^-l +Pin-^ +P2n-1 = «P2» +Pin-i =Pm-1 < therefore, by induotion, the result follows. Similarly we may shew that 9. We have P2M-l = '^Pin+P2n-l< B.nd p^^^bp^n.^+p^n-a, •"■ i'2n+l=(»* + l)i'2n-l + «P2n-2; and i>2„-i=ai'2n-2+i'2n-3". whence, hy substitution, P!in+i={'^^ + ^) Pia-i- Pans- Similarly we may shew that2P2„=(a6 + 2)2P2n-2-^2»-4'> .-. generally p„=(a6 + 2)2i„_2-jj„_4. all 10. The first expression = ax^ + ax2+ x^-h JJ ' 100 ' thus 775x100-711x109=1, and 775a; - 711)/ = 1 ; .-. 775 {X - 100) = 711 (2/ - 109) ; hence ^^ = ^^ =t, a;=711J + 100, !/ = 775{ + 109. 2. The convergents *o 455 ^"^^ 1 > 7 > gl 5 thus 455x73-519x64= -1, and 455a; - 519?/ = 1 ; .-. 455 (a; + 73) = 519 (2/ + 64) ; hence a; + 73 = 519t, 3/ + 64 = 455J. 3. The convergents *° 393 "^^ j , y , — ; thus 436 X 64 - 393 x 71 = 1, and therefore 436 x 320 - 393 x 355 = 5 ; whence 436 (a; -320) = 393 (2/ -355); .-. a; - 320 = 393f, y-355 = 436t. 4. Let x,y\>6 the number of florins and half-crowns respectively ; then ix + Sy = 79. One solution is a;=l, y = 15; thus the general solution is a; = l + 5i, 2/ = 15-4t. Here t can have the values 0, 1, 2, 8 ; hence there are 4 ways. XXVI.] INDETERMINATE EQUATIONS. 131 5. By trial x=l, y = 7S is a solution; hence the general solution is x = l + 15t, y = 78-llt; and S can have the values 0, 1, 2, 7; thus there are 8 solutions. 6. Let X, y he the numerators; then 7 + 5 = fnT> °^ 93i: + 7y = 73, the only solution of which is x=5, y = i. 7. Let X, y be the numerators; then T5~| = nT i ^^^^ i^ 2x-iy = l, or 2x-3y = -l. (i) The general solution of 2x-Sy = l is x = 3t + 2, y = 2t + l, and since j/<8, the values of t are restricted to 0, 1, 2, 3. Thus x = 2, 5, 8, 11; 2/ = l, 3, 5, 7. (ii) The general solution of 2x-3y=: -1 is re = Si + 1, y = 2t + l; thus a: = l, 4, 7, 10; y = l,3,5,7. 8. X pounds y shillings is equivalent to 20x + y shillings; hence 20x + y = -{20y + x); that is, 39x = 18y, or 13x = 6y. The general solution is a; = 6t, j/ = 13t; and as x, y are both restricted to values less than 20, it follows that t can only have the value 1; thus x = 6, 2/ = 13. 9. Eliminating z, we have 40x + 37y = 656. By trial, one solution is y — 8,x = 9; hence the general solution is a; = 9 + 37f, 2/ = 8-40t; thus t can only have the value 0, and a; =9, y = S is the only solution. By substitution we find z = 3, 10. Eliminating x, we have 4^/ + 7^ = 73; the general solution is y=13~7t, z=3 + it. Thus t can only have the values and 1. When f = 0, y = 13, 2 = 3, but the value of X is fractional; when t=l, y=6, z = 7, x = 5. 11. The general solution of 3i/ + 42=34 is 2/ = 10-4f, 2=l + 3t. Thus y = 10,e, 2; 2 = 1,4,7. From the equation 20a; -211/ = 38, we see that when ^ = 10, or j/ = 6, the value of a; is fractional; and when y = 2, x = i, z = 7. 12. The general solution of 13a; + 112 = 103 is a! = 2 + llj, z = 7-13t; thus a! = 2, 2 = 7 is the only solution. From 72 -By — i, we have y=d. 13. Put 2=1, then 7a; + 4?/ =65; thesolutionsarea; = 3,2/ = llJ x = 7,j/=4. Put 2=2, then 7x + 4y = iG; here the solutions are x = 6,y=:l}x = 2,y = 8. Put 2=3, then 7x + iy = 27; here a; = 1, y = 5is the only solution. Put 2 = 4, then 7a; + 4?/ = 8, which has no integral solution. 9—2 132 INDETERMINATE EQUATIONS. [CHAP. 14. Put X = 1, then 171/ + II2 = 107 ; solution y = 5, z = 2 put a;=2, then 17j/ + llz= 84; solution 2/=3, 2=3 put a; = 3, then 17y + ll2= 61, solution 2/=l, 2 = 4 put X = 4, then 17^ + 112= 88 ; no solution ; pntx = 5, then 17)/ + 112= 15; no solution. 15. Let N denote the number, x, y, 2 the quotients when N is divided by 5, 7, 8 respectively; then N=5x + Z = ly + 2 = %z + 5; hence Tj/- 5a; =1 and ly-Sz = Z. The general solution of 72/ - 5a; = 1 is x = 4 + 7s, 2/ = 3 + 5s. Substituting this value of 2/ in 72/ - 8z = 3, we have 35s - 82 = - 18, the general solution of which is «=2 + 8t, 2 = ll + 35t. Substituting for s, we obtain x = 56£ + 18, 2/ = 40J + 13, 2 = 35t + ll, ;^=280J + 93. 16. With the notation of the preceding example, we have W=3x + l = 72/ + 6 = ll2 + 5; hence 3x - 7!/ = 5, Bx-ll2=4. Thus X = 4 + 7s, y = 1 + 3s, and substituting for x, we have II2 - 21s = 8 ; whence 2 = 16 + 21i, s = 8 + ll«; thus x = 77S + 60, 2/ = 33« + 25, 2 = 21t + 16, iV=231t + 181. By putting t = 0, t= 1, we find that the two smallest values of N are 181 and 412. 17. In the septenary scale let the number be denoted by xOy ; then in the nonary scale it is denoted by s/Ox. In the septenary scale xOj/ represents the denary number y + .7 + x.V, or 2/ + 49x. Similarly in the nonary scale 2/Ox represents x + 81?/ ; hence 3/ + 49x=x + 8l2/, or 3x = 5y. The general solution of this equation is x = 5t, y = 3t; but x and y are both less than 7 ; hence x = 5, ^ = 3 is the only solution. Thus y + 49x, the value of the number in the denary scale, is equal to 248. 18. By hypothesis - = 5 + 5- ; hence 6 = — . By ascribing to a the values 1, 2, 3 11, we get the corresponding values of 6. 19. Since 250 and 243 have no common factor, no two divisions will be coincident. If a is the length of the two rods, then the distance from the zero end of the x"" division of the first is ^r^ , and of the 2/"' division of the second is ^^ . Hence the distance between these divisions is XXVI.] INDETERMINATE EQUATIONS. 133 ( X y \ 243a:~250y V250 ~ 243; "• "'^ 250.243 "' As the numerator cannot be equal to zero, tMs fraction will be least when 243x-2502/=±l. 250 1 07 The penultimate convergent to rj^ is ■^. , and 243 x 107 - 250 x 104 = 1 ; also 243 (250 - 107) - 250 (243 - 104) = - 1 ; thus the values of x are 107, 143 ; and the values of y are 104, 139. 20. Let a.', y, z denote the required number of times; then the three bells tolled for 23a;, 29y, Siz times, excluding the first of each. Hence 29?/ = 23x + 39, 34z = 23a; + 40 ; therefore Siz - 29?/ = 1. The general solution of this equation isi! = 6 + 29S; y = l + Sit. Now since the bells cease in less than 20 minutes, 29y, or 203 + 29 x 34i5, 997 must be less than 1200: that is t 'c^-^ — ;rr<2. 29 X 34 When (=0, y = 7, but the value of x is not integral; when J = l, )/ = 41, a;=50, 2=35. 21. Let o, 6 be a solution of the equation 7x + 9y = c, and let a be the smallest value of x for any particular value of c, so that 6 is the greatest value of y ; then the general solution is x = Qt + a, y = b-7t. Since there are to be 6 solutions, t is restricted to the values 0, 1, 2, 3, 4, 5. Also c = 7a + 96, and will therefore have its greatest value when a and 6 have their greatest values. Now b-7t is a positive integer ; hence 6 > 7i ; thus 6>35 ; and the greatest value of 6 is 41, for if 6 = 42, then t=6 would be an admissible value. The greatest value of a is 8, for if a =9, then t= - 1 would be an admissible value ; thus c = (7 x 8) + (9 x 41) = 425. 22. As in the preceding example, a;=llt + a, 2/ = 6-14*; where t may have the values 0, 1, 2, 3, 4. Thus the greatest value of a is 10, and since 6 must be greater than 4 x 14 and less than 5 x 14, the greatest value of 6 is 69; hence c = 14a + 116 = 14x 10 + 11x69 = 899. 23. The general solution of 19x + liy = c is a; = a + 14«, y = b-19t; where t may have the values 0, 1, 2, 3, 4, 5. Since zero solutions are inadmissible, a must lie between 1 and 13, and 5 must be greater than 5 x 19 and less than 6 x 19. Now c = 19a + 146, and is greatest when o=13 and 6 = 113, in which case c = 1829 ; also c has its least value when a = 1, 6 = 96, in which case c = 1363. 24. Let x = h,y=hhea, particular solution of ax + by = c, and Itt h be the smallest value that x can have for any particular value of c, so that k is the greatest value of y; then the general solution is x=h + bt, y = k-at, where { is restricted to the values 0, 1, 2, (n- 1). Since zero solutions are inadmissible, h must lie between 1 and 6-1, while k must lie between lH-a(«-l) and a-l + a(n— 1). Now c = ah + blc, and the greatest values of h and i> are 6-1 and a - 1 + a {» - 1) respectively ; hence the greatest value of c = (n + 1) 06 - a - 6. 134 RECURRING CONTINUED FRACTIONS. [CHAP. The least values of h and k are 1 and 1 + a (ra - 1) respectively ; hence the least value of c = (n-l)ab + a + i. This Example includes Examples 21 — 23 as particular cases. EXAMPLES. XXVII. a. Page 294. 1. VB=iW3-i=i+^; 4^=1^^'=! + ;^; .■. the continued fraction = 1 -1 — : 1+ 2+ 1+ 2+ ' . J ii, i ,„ 5 7 19 26 and the convergents are 1,2, -, -, —-, -— , o 4 11 15 2. ^/5 = 2+^/5-2 = 2+-^^; ^5 + 2=4+^5-2; .-. the continued fraction = 2 + ; . . . . : 4+ 4 + , ,, , 2 9 38 161 682 2889 and the convergents are =- > -r , ir- ■ — . — , ^ 1' 4' 17' 72 ' 305' 1292' 3. ^6 = 2 + ^6-2 = 2+-^; ^6 + 2 = 2+^2 = 2+-^; ^6 + 2 = 4 + ^6-2; .". the continued fraction = 2 H -: 2+ 4+ 2+ 4+ ' J ., . 2 5 22 49 218 485 and the convergents are - , - , — - , -— , — , — ^ 1 2' 9 ' 20' 89 ' 19a' 4. V8 = 2 + V8-2 = 2+-^-3L,; ^ = 1+ ^l-^^l^ _J_, ^8 + 2=4 + ;^8-2; .•. the continued fraction = 2 H — 1+ 4+ 1+ 4+ ' J .1, , 2 3 14 17 82 99 and the convergents are J , j, — , _, — , — 5. .m = 3 + VU-3 = B + -^; ^3^3+^/-ll-=3+^3; ;^ll + 3 = 6+Vll-3; XXVII.J RECXJERING CONTINUED FRACTIONS; 135 .•. the continued fraction=3 + T; — ^ — ;; — - — ; 3+ 6+ 3+ 6 + . i. ^ 3 10 63 199 1257 3970 andtheoonvergentsarej, _, -, _, _, ^^ 6. ^13 = 3^13-3 = 3^-^3; "^^ =' ^ "^ =' ^ ji^V V13 + l _. V13-2 3 . n/13 + 2 _, , J13-l _, . 4 3 -'■^ 3 -^ + ^13 + 2" 3 ~^"*'~3~-^+,7l3Tl' n/13 + 1 _, , V13-3 _ 1 . .•. the continued fraotion=3 + x/13H 1 1 1 1 I 1+ 1+ 1+ 1+ 6 + ' , ^, , 3 4 7 11 18 119 and the oonvergents are ^ , j, -, -3-, — , — , 7. V14 = 3WU-3=3.,^,; ^/ii±-^ = l..^=l. 2 ^14 + 3' 5 '5 ~ ^^14 + 2* 2 "^2 ^^14 + 2' 5 " '^ 5 ~ "^v'14 + 3' ^/14 + 3 = 6 + V14-3; .•. the continued fraotion=3H- = — ^ — = — j: — : 1+ 2+ 1+ 6 + 3 4 11 15 101 116 and the oonvergents are ^ , j, -j' -|-> 27' "31 • 6 ^22 + 4 ^^ ^22-2 ^^ 3 n/22 + 2 _o. n/22-4 _„. 2 8. V22=4+^22-4=4+-^^^^ = 2+ '^ „ =2+- 3 ^3 ^^22 + 4 n/22+4^„^ V22-2 _^^ 6 3 3 V22 + 2 ;^22 + 4=8+;^22-4; ^/22 + 2 ^22 + 4 ^^^ ^22-4 ^^^ 3 2 ^2 ^^22 + 4 e ^6 "^/22+4 .-. the continued fraction=4 + jL ±±-±±-J-. 4 5 14 61 186 197 and the oonvergents are J, j. -3-, jg, gg- 42 • • 9. 2^3=V12=3W12-3=3 + -^; t/1^^2^^ = 2+ 1 x/12 + 3' ^12 + 3=6+^/12-3; 136 EECURRING CONTINUED FRACTIONS. [CHAP. /. the continued fraction=3 + 5-— ^ 5— g— ; ,^, ^ 3 7 45 97 627 1351 anatheconyergentsajrej, ^, ^, ^, jgj, j^, 10. V32=5W32-5 = 5+-^; ^^ = 1+^ = 1 + -^; ^32+2 _, , V32-2 _, , 7_ ^32 + 2_ V32-5 ^ 1 ~i -^ + ~"i --^+^32 + 2' 7 ~ 7 ■^"*'V32+5' ^32 + 5 = 10 + ^32-5; .•. the continued fraction=5+ r— yt Y+ loT ' 5 6 11 17 181 198 and the CQUYergents are J- ; =-, -^. -g- , -^ , -^ , ^45 + 5 _„ V45-3 9 . ^45 + 3 ..V45-6_ 1 . ^i~-'' + ~r~'-''"^V45 + 3' 9 ~ 9 ""^^45 + 6' ^'45 + 6=12 + ^45-6; .-. the continued £raction=6 + j^ _L _L _ __ _. ; 6 7 20 47 114 161 and the convergents are J , j, -g-, y, -j=- , -^, 12. V160=12+V160-12 = 12 + -^; ^3^^^^0-l V10 + 4_ 4V3.0-5 _ 15 V10 + 5_ VIO-IO 9 ~ ■*■ 9 ~ ■^4;,yi0 + 5' 15 ~ + 15 2^/10 + 5 _g, 2V10-5 _L 15 4^10 + 10 ., 4^10-5 . - _D+ 2 -"■''iVlO + lO' 15 -^'^ 15 9 ~ "*'4^10 + 4' 2 ~ "'■2^10 + 5' = 1+— ?— ■ ^+4v'10 + 5' XXVII.] RECUEEING CONTINUED FRACTIONS. 137 4^10 + S ^ i ^v/10-4 _. 4 yiQ + l ^10- 3 9 ' "^ 9 ~ "^VlO + l' 4 ~ ■''"i = ^ + 4(V10+3) = 4(V10 + 3) = 24 + 4V10-12; /. the continued fraction = 12 + ^; — = — :; — ^ — = — = ;^ : 1+ 1+ 1+ 5+ 1+ 1+ 1+ 24 + , ^, .12 13 25 38 215 253 and the oonvergents are Y , y, y y ^y 20' 13. V-=.W2X-.=..^,; ^^=1.^^=1+-^^. V21 + 1 V21-3 3 . x/21 + 3 _g, V21-3 _ 4 V21 + 3 _ ^/21-1 T , 5 . v/21 + 1 , , x/21-4 , , 1 ^^-^ + ~^~-^+;/2m' ^5— =^ + — 5~=^+^72lT4' ^21 + 4=8 + ^21-4; .•. the continued fraction = . — 7 — ;; — ;r — = — -= — ^ — ; 4+ 1+ 1+ 2+ 1+ 1+ 8 + 112 5 7 12 andtheconvergeutsare J, g, g, ^, ^, ■^, ; 14. x/B3 = 5 + VB3-5-5+-^;^^ = l + V3i^«=l+^; V33 + 3 _„, V33-3 „, 8 . ^33 + 3 J33-5 1 ""3 ^+~3~-'' + V33 + 3' ~8~~^ + ~8~-^+V33 + 5' v/B3 + 5 = 10+V33-5; .■. the continued fraction = ^ — f — = — ^ — =7^ — ; 5+ 1+ 2+ 1+ 10 + 1 1 3 4 43 219 andtheconvergentsareg, g, j;^, ^g. 247' 1258' 15- ^/^^°=l+^'=l+;A^=^/^«+^=^ow3o-5 \/30 + 5 _„, x/30-5 . ~5 -^+"-5~' .•. the continued fraction = l + :j7r — h— .. 1U+ ^ + , 1 11 23 241 505 5291 andtheconvergentsarej, ^, ^, ^^q. Jgl' 4830' ='"-^;^5' 138 RECURRING CONTINUED FRACTIONS. [CHAP. 1ft /7_ 7 . „„, v/77 ,^v/77-7 ,, 4 v'77 + 7 _a , ^/77-5 _^ , 13 J77 + 5 ,,^77-8 . 1 4 -^+-^--^+^77T5'T3 — ^+~T3-=^+;mT8 = v/77 + 5 _, , V77-7 _^ , 7 x/77 + 7 „^s/77-7 4 -^+^r~-^+V7r;7' ~7^-^+^r~= .■. the continued fraction = - — -- — - — —- — , 1+ 3+ 1+ 16+ 1+ 8+ 2+ 3+ 1+ ' , , , . 1 3 4 67 71 280 andtheconvergentsarej, j, g, -, _, — , 17. ^17=4+V17-4 = 4 + -^ji^; ^VJ + i = % + JVJ-i; .: the continued fraction =4 + ;- ... .; 8+ 8+ ' ,^, ,4 33 268 2177 and tne convergents are :j- , — , -^ , -r^ , 1 o 00 DJo .-. the error in taking -pr=- is less than ^rrrrx and greater than 65 (65)2 o 2(528)2 18. \/23 = 4+— - r— =— 5— ; and the convergents are 1+ 0+ ±4- + 4 5 19 24 211 235 916 1151 1' 1' 4' 5' 44' 49' 191' "240 ' .-. the error in taking t^ is less than and greater than ■ 191 (191)2 """ ^"""" ""•" 2(240p- 19. \/101 = 10 + 2jj— 2Q— ; and the convergents are 10 201 4030 1 ' 20 ' 401 ' The third convergent differs from ,^101 by less than ,,^,,. , and is therefore (401)2 correct to five places of decimals. 20. \/15 = 3 + J— g— ; and the convergents are 3 4 27 31 213 244 1677 1' 1' 7 ' 8 ' 55 ' 63 ' 433"' The seventh convergent differs from ^15 by less than tt — r,; , and is there- (433)2 fore correct to five places of decimals. xxvilJ eecuering continued fractions. 139 21. The positive root of a;^ + 2x - 1 = is ;^2 - 1. Now ^2-1=-^; ^2 + 1 = 2 + ^2-1; 111 .', the continued fraction =-; 2+ 2+ 2 + 22. The positive root of x^ - 4a; - 3 = is ^7 + 2. Now .^7 + 2 = 4+^7-2 = 4+—^^-; ^1=1 + ^=1+-^; ^7 + 2 = 4 + ^7-2; .'. the continued fraotion=4 + :; — :; — = — -; — 1+ 1+1+4 + 23. The positive root of 7a;2 - 8a; - 3 = is "^^^"""^ . -. s/37 + 4 ,^37-3 ^ , 4 ^37 + 3 _ ^37-5 _ 3 . ^/37 + 5 _ ^37-4 , , 7 .•. the continued fraction = 1 + - — — - — 2+ 3+ 1 + 24. The roots of a'' - 5x + 3 = are ^'^ . ^^ 5 + ^13 ._,\/13-3 , , 2 ^13 + 3 ^13-3 o, 2 Now-|-=4+^4— =4+-^j^3;V__^3+V__ = 3+__. /. the continued fraction = 4 + -r— - — 3+ 3 + . . 5-^13 6 v/13 + 5 . ^13-1 . 2 Again, _^_=^-p-^; — 6— = 1+ "6— = l+;/l3Tl' ^13 + 1 „, V13-3 _, , 2 . \/13 + 3_o^s/13-3 .•. the continued fraction = — ;-— - — - — 1+ 1+ 6+ + 25. Letii; = 3+,-; — ;; — ; thena;-3=c-7 gj , whence a; =^'10. 6+0+ b+(x-6) ^ 140 RECURRING CONTINUED FRACTIONS. [CHAF. noT^ 1111 ^^ 11 26. Leta: = , — - — ■- — - — :theiia; = = — . 1+ 3+ 1+ 3+ ' 1+ S + x 3 + aj Therefore a; = J , or a;'' + 3x-3 = 0; and the given continued fraction is the positive root of this quadratic. «r,TT ,111 „la; 2x + l 27. Herea; = 3+; — k— 7, — ; t^; ox x-3 = z — t^ ?- = k vi 1+ 2+ 3 + (a:-3) ' 1+ 2a; + l 3a; + l whence we obtain 3x^ - lOx -4 = 0, and the continued fraction is the positive root of this quadratic. 28. Here a; - 5 = = — = — :; -; which reduces to 3x^ = 96. There- 1+ 1+ 1+ x + 5' fore x=4:J2. 29. By the method of the preceding examples 11 „- J 1 1 1 /5 ^+1^ 6T =^^'' ""^ ^+3T 2T = Vs' whence the required result follows. Or it may be proved thus: 3(i + 3T2T )=^+3T2T =^+iT6T on rrv, • li -9 + x/145 , -Il+Ji45 30. The expressions are equal to j^ , and ~ . The difference of these values = - . EXAMPLES. XXVII. b. Page 301. 1. Ja' + l = a-}:{Ja' + l-a) = a+ ; ^0.^+1 + a Ja^ + l+a = 2a + {Ja^ + l-a) = Thus Ja^+l = a + 5-— ;-— - , ^ 2a+ 2a + and the convergents are -, —^ , .„,., . —5-5 — . Ja^~a+a~l ^^ ^ Ja^-a-{a-l) _,, ^ Jo' -a +a- 1 ' ''-I J'cfl^-\-a-l' sja' -a+a-1 ., , i-r, =-\ -i^^ j — . = 2 (a - 1) + {^a' -a-a-l)= XXVII.] EECURRING CONTINUED FRACTIONS. 141 Thus VS^=(„_l) + _L__i_ „, , o-l 2a-l ia'-Sa + l 8a^-8a + l The oonvergents are -y- . -g- . 4^,3 ■ sa-i • 3. v'a»-l=a-l+Va2-l-(a-l) = (o-l) + -7=;== ; V""-l +a-l 2a-2 "^ 2(1-2 ^a'^i^+a-l' N/^ + — ;i ^ — > — s — • ° 1 1 2a - 1 2a \' a a a Jd'+a + a 1 1 v/a^ + a+a Ja' + a+a Ja^ + a-a =^ + - a a Thus /v/ l + -=l + s— r;rr o— ToT A' a 2a+2+2o+2 + ,1 2a + l 4a + 3 Sa^ + SaH- and the convergents are - , -^^ , ^^^ . -g^q:^ / , 2a. Ja^b^ + 2ab , JaV + 2 ah -ab 2a = a + Ja%^+2ab+ab' Ja'b^ + 2ab+ab , Jo^P+2a6-a6 6 . 2L = (3+— ^ ^O-h . ) 2a 2a ^a2i^+2a6+a6 Ja^b'' + ab + ab_ , Ja? + V-ab _ 6 ^"'^ b - /T~2i 1111 Thus V" +T="+6+2^6^2-^ ' a ab + 1 2a=6 + 3a 2a^b^ + iab + l and the oonvergents are J, — g- , gai + l ' 2al^ + 2b ' 142 .RECURRING CONTINUED FRACTIONS. [CHAP. - / , a Jn^n^ -an , Ja^n^ -an — la-l)n 6. \/ a'-- = —- =a-l+- i '— \ n n n 2an -a-n = n-T-i- ; ^JaH^-an + {a-l)n tja-ii^ - an + {a — l)n_ sjd'ii^ — an — (an - a) _ , a 2an-a-n ~ 2an-a-n Ja;^^-an+an-a' Ja'^n^ -an+an-a „ , , , Ja^n^ -an- (an - a) — = 2 (ra - 1) + -5^ ' a ^ ' a „, _ 2an-a-n =2(ra-l)+ ; ija^n^ -an + an-a ^a'n? ~ an + an - a _ JaV -an- (an -li) _-. n lan-a-n ~ 2an-a-n ~ ^a^n'-an +an-n' ija^n^ -an + an-n _ . . ^a-n^ -an -(a-l)n J I a — ij + — . . , , n n Thus ./a^-- = (a-l) + ~ --^ ^ i ; V n "■ ' 1+ 2(n-l)+ 1+ 2(a-l)+ ' J , , , a-1 a 2an -a-1 2ara-l and the eonvergents are — 7— , :r 1 — s = — ■ . 1 ' 1' 2ij-I ' 2re 7. V9a' + 3=3a+(V9a' + 3 -3a) = 3a+ — x/9a2+3+3a' V9a2 + 3+3a JgaF+3-3a „ 1 5 — ^a + -=2a+ ; ** "* V9a2 + 3+3a J = 6a + (V9aH3 -3ffl) = .... Thus ^M+3=da + ^~ ~Jl-...; 2a+ 6a+ 2a+ 6a+ ' and the oonvergents are 3a 6a° + l 36a3+9a 12a'^ + 2ia' + l 432a« + ISOa^ + 15a 1' 2a ' 12a2 + l ' 2ia^ + ia ' 144aH36aHl ' 2 8. Weha-ve x=p + - n\ i.-\-y ^ ■'* where «= /o-v Prom (2), (l + 2/)(l-W)=2/. ^rom(l). (l + y)(x-p) = 2 (3); ••• 2(l-p»/)-j/(a;-p) (4). XXVII.] EECUERING CONTINUED FRACTIONS. 143 From (3), l + y= ; from (4), «=-^. x-p' ^ " " x+p By subtraction, 1 = = -5-^ ; whence x'=p^ + ip. x-p x+p x^-j? " '■ 9. i* ( "1+ ]^ |=pai + ' ^ ]^ =i'«i+ 5-; where jB„=a3 + Similarly pBj =p la^ + - \ = iiag + ; and so on. 10. From Ex. 1 we see that the complete quotient at any stage is always ^a^ + 1 + a; hence, as in Art. 358, /;;2TT- ( \/a^ + 1 + a) P« +Pn-T. Multiplying up, and equating rational and irrational parts, (a^+l) °'^ Pn+l-Pn-l=Pn'^ •■■PS+P6+P7+-+ Pan-i = iPi -Pi) + (Pa -Pd + (Pa-Pe)+- + (Pm - P'.n-i) =P^-Pi- Similarly for the other result. ,„ T ^ 111111 17. Let x= — , -T ... : ' a+ 6+ c+ a+ b+ c + 111 6(c + .r) + l then X = -. = 7—; — Vw % » a+ b+ c + x {ab + l)(c + x) + a on reduction we obtain {1 + ab) x^ + {abc + a -b + c)x- {be + 1) = 0. Denoting the value of the second continued fraction by y, we have by inter- changing a and 6, (1 + ab) y^ + (abc -a + b + c)y-(ae + l) = 0. Subtracting and rearranging, we have {l+ab)(x^-y-) + (ab + l)c{x-y) + (a-b)(x + y) + c{i-b) = (i; that is, (l + ab){x-y)(x + y + c) + (a-b)(x + y + c) = 0\ now x + y + c is positive, hence {l + ab)(x-\j) + {a-b) = Q; which proves the result. 18. In Art. 364, we have proved that the 27t"' convergent J(Pn NqA_ p,-' + Nq„ \ 2V3« P„J Spn^n ' and this we denote by ^--- ; hence San S2n=2i'n9'n. B.nd p^ = p„^ + Nq,,". Also from Art. 3C4, <'lPv,+Pn-l = ^9n' 'h .•.J'n=-^?„==M«-i-i'n-i=^^■, then x^-z^ = 3y{x-y). Put m{x + z) = 3ny, n[x-z) = m{x-y); then by cross multiplication X y z '6n^-m? ~m!' + 2mn m- - 'imii + iijt' ' 16. We have {x + y)^=z'-y\ Put vi{x + y) = n{z + y); n(x + y) = m{z-y); then '^ =^^ =_-_ . -vi' + ^mn+ii' vi'-v? »«-'+«" 17. We have Sx^ = 2= - j/'. I'ut 5mx = n(z + y)\ nx=m(z — y); then -^-= y ^ ' . 2mn 5m^-n^ bm^ + n' 18. If X and 2/ represent the two numbers, x'' - y^ = 105. The factors of 105 are 1, 105; 3, 35; 5, 21; 7, 15; the solution may easily be completed as in Art. 377. 19. Denote the lengths of the two sides and hypotenuse by x, y, z respectively; then x'' + y- = z^, or x'''=z^-y''-. Put mx=n(z + y), and nx = m[z-'y); ii X y z 2m7i ■m^ — n- m-+n- 20. Let X, y be the integers; then x^ + xi/ + !/- = perfect square=2^ say; thus x {x + tj) t= z- - y'^. Put mx=n{z + y), n(x + y)=m(z-y); then 2mri + w' m^-n" m^ + nm + n"' 21, Let X denote the number of hogs bought by any one of the men, then since x shillings is the price of each hog, x^ shillings represents the value of the hogs bought by this man. Similarly if y'^ shillings be taken to represent the value of the hogs bought by the wife of this man, we Ijave x''-y^=6S. Proceeding as in Art. 377, we find for the solution x = 32, 12, 8; y = Sl, 9, 1. Thus the men bought 32, 12, 8 hogs, and the women 31, 9, 1. Further, Hendriek bought 23 more than Catriin; thus Hendriek bought 32 hogs, and Catriin 9 hogs; also Claas bought 11 more than G-eertruij ; thus Claas bought 12 and Geertruij 1 hog. Hence Cornelius bought 8, and Anna 31 hogs ; therefore we have the following arrangement Hendriek 32) Claas 121 Cornelius 8) Anna 31/ Catriin 9) Geertruij Ij * 150 INDETERMINATE EQUATIONS. [CHAPS. 22. The sum of the first n natural numbers is ^5i!L--i. This ex- pression is a perfect square when n=k^, provided that k^ + 1 — 9— = a perfect square=a;'' say; that is, k^-2x^= -1. Since J2=l + i: ... , the number of quotients in the period is odd, ^2+2 + and the values of k are the numerators of the odd convergents. [Art. 370.] Again, the expression — ^r — - is a perfect square v? hen m + 1 = k'^, pro- jc'2 _ 1 vided that — ^ — = a perfect square = 1^ say; that is, k'^- 2x^ = 1. In this case the values of k' are the numerators of the even convergents. [Art. 369.] EXAMPLES. XXIX. a. Pages 321, 322. TT / ,v, „i ,„ n(n + l)ln+2)(n + 3) „ 1. HereM„=n(n + l)(« + 2), and S„=— 5 '^ m ^^p. 1, T fi 1 rr n ii, cr n(n + l){n + 2){n + 3) when n = l, we find C=0; thus5„=— ^ -^ — — ' . 3. Here M„ = {3n-2)(3n + l)(3n + 4), „ _ _ , (3)r-2)(3n + l)(3n + 4)(3« + 7) ••■ S„-C + ^-^ ; 1 ■, 1, nn ^ 1-4.7.10 ^, ^ 14 when 11 = 1, we have 28 = Ch =-;; ; thusC=— -. .: S„ = j^{3n-2){3n + l){Sn + 4){3n + 7} + ^^. 4. Here i(„=n(re + 3) (n + 6) = n(7i+l) (?i + 2) + 6ra(ra + l) + 10«. .•.g„=(;+ "(" + ^)'" + ^)<» + ^) + 2»(. + l)(„ + 2) + 5n(>^ + l); when n=l, we have 28 = C + 6 + 12 + 10; thus (7=0; ,. ^„3='iMK + 5n + 6 + 8» + 16 + 20} = '"" + ^)(" + ^'''""^' 5. Here «„=n(n + 4) (?i + 8)=n(n+l)(«+2) + 9n(n + l) + 21ji; .-. S,.=(7 + «J^i±HMi!i±i) + 3«(„+l)(n + 2)+ ?ili^) . XXVIII, XXIX.] SUMMATION OF SERIES. 151 and by putting n=l, vie find (7=0; , ,, iln + 2)(n + S) „, „. 211 .-. S =n(n + l)|^ ^ -'+3{n + 2) + -^j = ^^i^^(K'' + 17n + 72)=|n(n + l)(n + 8)(n + 9). 6. By Art. 3S6, S„=C =■ , and it will be found that C = 1 ; .'. Sn= 7 ; and clearly yS„ = 1. 71+1 7. neres(„=77i ttt-s tt ; and S'„=C-;r-77r — . ' " (3;i - 2) (,3)1 + 1) 3(3)1 + 1) Put w=l, tlien-=-g— j+C. thusC=-, 'S'„ = g— -j, and '5'»=y- 8. Here «,. = ,3„,_ , wo,,^]wo,,^o. ; and S„ = C - (2)s-l)(2)i + l)(2)i + 3)' " 4(2)t + l)(2)i + 3) Put 71 = 1, then rr=C- 16 4.3.5 •'• C'=TTT. and iS'„=:r-r 12' "12 4(2)1 + 1) (2)j + lj 9. Here "» = (3)i - 2) (3)i + 1) (3)i + 4) ' 1 .-. S„=C- 6(3)l+l)(37^+4)' "■ _ n+3 _ 1 3 10. Here «»-„(„ + !) („ + 2)~ (71 + 1) (n + 2) "''n(7i + l)()i + 2)' « + 2 2()i + lJ()i + 2) 5 •when )7 = 1, we find C= j. 5 1 3 _ 5 _ 2)1 + 5 ■■■^"-4-„ + 2 2(7i + lj(?i + 2)~4 2(7i + l)(n + 2)' n_ (n + 2)-2 11. Here m„- („.^2)(n + 3)(7i + 4) " (ra + 2)(n + 3)(7i + 4) 1 2 "()t + 3)(n + 4) (71 + 2) (71 + 3) (71 + 4) ; &0. 152 SUMMATION OF SERIES. [CHAP. 12. ^^'^^'^»= „^,^ + i^n + .^^ = (n+l){n + 2)~ n(,i + l)(n + -4)' 13. Here v„ = n{n + l){n + 2){n + l) =«(n + l)(n + 2)(« + 3)-2n(re + l)(n + 2); .-. S„=C-pn(n + l)()i + 2)(m + 3)(n + 4)-in(n + l)(™ + 2)(n + 3); &e. 14. Here S„=n''(l + 2 + 3+... + n)-(l' + 2s + 3'+. ..+«') 1 ,, ,, («(m + l)l ^ 1 o, 2 n = ^ "■' (» + 1) - I 2~ J " i ^^" " '■ 15. Here w„ = (ji-l)n(n + l)ri = (n-l))s(n + l)(n + 2)-2(n-l)n(n + l); &c. 16. Here u„={n + l){n + i) {{n + 2){n + 3) + 2\ = (n + l)(n + 2)(m + 3)(re + 4) + 2(B + l)(n + 2) + 4{ii + l); .-. S» = C+p(n + l)(n + 2)(n + 3)(7i + 4)(»+5) o + |(« + l)(n + 2)(n + 3) + 2(K + l)(r! + 2). When )i = 1, we find C= - 32, and S» reduces to p(K+l)(re + 2)(3n3 + B6«2 + 151n + 240)-32. _rf 4n2_4_n2_3 n° _«' 3 4n^ 17. Here«„-^ .^^^3_j-- ^ 4-4n2-l-4 16'4,t^-l 71= 3 3 4 16 16(4n^-l)' ?i( n+l)(2n+l) _ 3n _ ^ Jl _ 1 ) .-. S„- 24 16 16 I2 2(2)1 + 1)) ?((n + l)(27! + l) 3 inv' + in _m(n + l) j2)i + l 24 16 2(2rt + l) (2)1 + 1 _ 3 I \ 3 2)i + l]' ?)(n + l)(4«'' + 4)i-8) _ (ra-l)?i(w + l)()t + 2) ~ 24(2n + l) 6(2ii + l) ' 18. HereM„= „,„'-,, =«(n + l) n()j + l) ^ )i(ii + l)' «f)i + l)(» + 2) )( rj + 1 XXIX.J SUMMATION OF SERIES. 153 19. Here«„ = l!^±l)|^±M±^=„ + l+^lMl)__ n' + 2n n{n + l)(n + 2) =n + l + (n + l)(n + 2) n(ii + l){ii + -A)' _ »i(»i+l) 2 1 by putting n = l, we find G=^, a 20. Here^„= ^"' + " + ^)^"'-" + ^) = "° + " + ^ =l+ ^ • .■. S„= C+n — r ; and by putting re=l, we find C = l. lra+r-2 21. The m"" term of the r* order=r-^ ., . , and the ?■«'' term of the |ra-l r-1 lr+m-2 n* order = , — Ir-l |n-l ' (n + r-2 22. The n'" term of the r"- order= ' ., , , and the (n + 2)"' term |n + r-2 of the (r - 2)"' order = ,' , „ ; ' |m+l|r-3 ' .-. (j--l)(r-2)=?i(re + l); ■whencer-2=n. 23. The sum of the first n terms of the r* order of polygonal numbers =^7i(n + l){(r-2)(«-l) + 3}; [Art. 390.] 1 »•-' 1 .'. therequuedsum = g{n-l)m(?i+l) S (r-2)+5m(n + l)(r-l) =^(ra-l)n(n + l){r-2)()--l) + in(n+l)(r-l) = ^ -^-^ -' {rn - 2n - r + 8}. 12 154 SUMMATION OF SEEIES. [CHAP. EXAMPLES. XXIX. b. Pages 332, 333. 1. The successive orders of differences are i, 14, 30, 52, 80, 114,... 10, 16, 22, 28, 34,... 6, 6, 6, 6,... Assame u"=A + Bn + Cn'; -whence by putting for n the values 1, 2, 3 suc- cessively, we get i=A+ B+ C,. li=A + 2B + iC, 30=A + 3B + 9G; and from these equations we find ^=0, B = l, (7=3. .'. the to"" term = Sn' + n. .: the sum of n terms = 3Su^ + 2)i=r!(n + l)^. 2. We have 8, 26, 54, 92, 140,... 18, 28, 38, 48,... 10, 10, 10,... n lo, ■,^ 10(?l-l)(»-2) , „ „ .-. K„=8 + 18(n-l) + — ^ _!_v -' = 5n= + 3n; .-. Sn=5Xn' + 3Sn = ^n(n + l){5n + 7). 3. We have 2, 12, 36, 80, 150, 252,... 10, 24, 44, 70, 102,... 14, 20, 26, 32,... 6, 6, 6,... Assume v„=A+Bn+ Civ' + Dn^ ; then by the method of Art. 397, we find w„ = n' + n-. .: S„=2ji3 + 2n2=in(n + l)(ra + 2) (3)1 + 1). 4. We have 8, 16, 0, -64, -200, -432,... 8, -16, -64, -136, -232,,.. -24, -48, -72, -96,... -24, -24, -24,... XXIX.] SUMMATION OF SERIES. 155 .•■«„=8.f8(n-l)-^^'"-iH"-2)_ 2^("-l)("-^)("-3) \a 'O _ 24(re-l)(?i-2) f, ra-3) „ , , ,,, „, = 8n i — r^ '- jl+-g- j-=8n-4n(n-l)(it-2) = -4n2{?i-3); .-., by the method of Art. 396, we find S„= -n(n + l)(?i2-3!i-C). 5. We have 30, 144, 420, 960, 1890, 3360,... 114, 276, 540, 930, 1470,... 1G2, 264, 390, 540,... 102, 126, 150,... 24, 24,... .-. i(„ = 30 + 114(n-l) + 81()i-l)(n-2) + 17(n-l){re-2)(»-3) + (K-l)(»-2)(n-3)(n-4); that is, «„=7i« + 7n' + 14n* + 8n=n(n' + 7n2 + 14)i + 8) =M(n + l) {n^ + 6n + 8)=n(n + l) (n + 2) (n + 4). And, by Art. 383, »„= — n(n + I) {n + 2) (n + S) (4it + 21). 6. By the method of Art. 398, we have 1, 3, 7, 13, 21, 31,.., 2, 4, 6, 8, 10,... 2, 2, 2, 2,.., Thus the scale of relation is (1 - x)^. S=l + 3x + 7x^ + 13x'+21x'+... ~3xS^ -3x-^x^-21x^-39x*- ... 3x25= 3x^+ 9afi + 21x^+... -x^S- - icS- 3x>-... By addition, S {l-x)^ = l + x^; . Q-i±5i ■ (1-X)3" Examples 7, 8, 9 may be solved in the same way. 10. We have 1, 16, 81, 256, 625, 1296,... 15, 65, 175, 369, 671,... 60, 110, 194, 302,... 60, 84, 108,... 24, 21,... ,•. the scale of relation = (1 - x)'. 156 SUMMATION OF SERIES. Now S= 1 + 16a; + Sla;'! + 256^5 + 625ii;4 + 129Gx= + . . . -OxS = - 5x- 80a;2 - 405a;3 - 1280a« - 3125a;' - . . . Wx"-S = lOicH 160x3+ 810a;^ + 2560a;'+... -10x^S= - 10a;3- 160x*- 810a;'-... 5x since x=.--. (1 - x)» 4 a 12. Put x=-p, then we have 5 S = 1 - 4x + 9a;2 - 16a;5 + 25a;« - 36a;' + . Sx= x-Ax''+ 9x3-16a;< + 25a;'-. .-. S(l + x) = l-3x + 5a!2-7a;3 + 9x^-llx5+. „ 1-x 25 . 1 .•. S=j: r5 = TT. Since x=-. (1 + x)3 54 5 1-a; = (l + x)='' 13. We have 9, 16, 29, 64, 103,... 7, 13, 25, 49,... 6, 12, 24,... .•. , as in Art. 401, we assume u^=a . 2"-' + 6n + c, Put n=l, 2, 3 successively; thus we obtain a=&, 6 = 1, c = 2. .-. «„=3.2" + 7! + 2. .-. S„=6(2'.-l) + !LM + 2» = G(2»-l)+^i!^). XXIX.] SUMMATION OF SERIES. 157 14. We have 2, 12, 28, 50, 78,... 10, 16, 22, 28,... 6, 6, 6,... We may therefore assume u„=A+Bn+Gn^ + Dn'. And as in Art. 397, we find «„=n'-()H-l)'^ = n»-ji-(n2 + » + l) = (n-l)«{n + l}-)i()i + l)-l. Whence S„ is easily found. 15. We have 2, 5, 12, 31, 86,... 3, 7, 19, 55,... 4, 12, 36,... .•. M„ = a . 3"-^ + bn + c, and as in Art. 401 we obtain a = l, 6 = l,c = 0; .-. «„=3»-i + ™, and S„=l(3n_l) + 'ii±ti) = ?l±I5^±^zi. J J 2 16. We have 1, 0, 1, 8, 29, 80, 193,... -1, 1, 7, 21, 51, 113,... 2, 6, 14, 30, 62,... 4, 8, 16, 32,... .•. ]/„ = a . 2»-^ + bn? + cn + d; and as before we find a=4, 6= -1, c=-2, _ (ro-l)2''-' _ *" m(«+l) ~ra + l ?2 ' 71.2" ■'■ S'„— n + 1 ' H. A. K. 11 162 31. We have SUMMATION OF SERIFS. _re+3 _ 3 1 m(TO + 2)~2ra 2(m + 2)' [chap. ra + 3 " m{n + l)(n + 2)'3» 2K{n + l)'3»-i 2 (n + 1) (m + 2) ' 3»' .". Sn = -T " 4 2 (71 + 1) (71 + 2) 'S"" 32. The m"' term of the series 1, 5, 11, 19 is Ji^ + n-l. Now u.= 7i''+n-l_n (n + 2)-(n + l)_ n n + 1 |n + 2 ln + 2 In + l |?i + 2' '2 \n + 2 33. The n" term of the series 19, 28, 39, 52, is n^+Sn + l'i; ^^ jj2 + 6« + 12 ,6 2 /„ 6\ /, 2 \ """^ ^»-T2r=^+»"«T2=('+»j-(^+,r+-2) 2(» + 3) w + 4_ ji n + 'i' «2 + 6n + 12 1 1 + 3 1 ra + 4 " m(n + l)(ft + 2)"2»+i m(7i + l)'2'' (n + 1) (ii + 2) ' 2"+! ' •'. Sn = 1 " n + 4 1 (ji + 1) (n + 2)'2»+i' EXAMPLES. XXIX. c. Pages 338—340. 1. We have e='=l+a,+| + |+|+|+..., By subtraction, _x_-i ""^ '^ ^ ^' ^ -•^"'' + i2_~]3_+j4""l5_+-- e»=-e-«=2a; + 25. ^^IX-] SUMMATION OF SERIES. 163 2- ^-U-2J + (2--3-) + (3-4) + - -U + ¥ + F+-j-i(2 + 3+4+-) = -log{l-a;)--{-log(l-a!)-a;}=i^log(l-a;) + l. 3. By writing down the series for e'', «-*, e^ and e-'^ tlie result is easily obtained. 4. Here _ t \1 jr + TO-l lr + K-2 'r + ra-l' ■ (r-.o)„- & (^1 " + 1 U ^ 1^ " \r + n-2 \ r + n-lj |r + )t-2 ~ |r + H-l' Thus (r - 2) «,=^ - 1^; and (r - 2) 5= ^^ . 5. S = l + 2x + ^x^ + ^x' + -^x*+... (2 [3 |4_ 6. WehaveeJ"'=l+2)a;+^+...+V-^+ T^+-.. .■. S= coefficient of x' in e'"' x e'i" or e»+g)»- ^^ + '?)'" . 7, The given series may be expressed as the sum of the two series n n(n-l) 1 n(re-l)(n-2) 1 1 + na; J2^ ' (T+nxp "^ J3^ ' (l + ma;)^" "■ 11—2 164 SUMMATION OF SERIES. [CHAP. \ 1 + nxJ 1 + nx \ 1 + nxJ 8. The scale of relation of the recurring series l + 3a: + 5a;^ + ... is (1 - k)2. Art. 398. S„=l + 3x + 5x^ + ... + (2n.-l)x''-\ - 2xS„= - 2a; - 6a;2 - . . . - (4ra - 6) x»-i - {in - 2) a;", x'Sn= a;2+ ... + (2n- 5) a;"-! + (2n - 3) a;" + (2n - 1) a^'+i; .-. (l-a;)2 5'„=I + a;-(2n + l)a:'' + (2B-l)a;''+i = l + a!-a;"{(2n + I)-(2n-l)a;}. When ^ = 2^^3i. we have /.2„ _ i)2 '^» = 2^^^! = tliat is, S„=m(2»-1). (1 nln-l) „ n (n - I) (re - 2) ., + ar)"=l + na;+ ' ., ' a;^ + — rf^^ -x'+.... ^ li / 1V-1_?5 "("-!) 1 re(re-l)(ra-2) 1^ V xj "" a;"^ [2 •a;2 [3 •a;3 + -' .'. S=the term independent of x in (l + a;)" II — 1 = the term containing .r" in (a; + 1)" (a; - 1)", that is in (s^ - 1)" ; iSi=0, if n is odd; and S=' 2 Ire 5 ^^^~ I - X)^, if n is even. n ^ 10. The given series is the sum of the two series e^'e,^ - 1 and e'"''s«' - 1. Now W=e'°e<*, therefore e>'>e.2_2^ and e2iog,2_giog,4_4. ti^g «= (2-1) + (4-1) = 4. XXIX.J SUMMATION OF SERIES. 165 ■'■^" "" (2B-rl)2n(2n+l)~2 V2re-l~2ra"''2n + lj' „„ /I 2 1\ /'I 2 1\ /I 2 1\ /I 2 1\ •■•2^=(l-2 + 3)+ (3-4 + 5) + (5-6 + 7) + (7-8 + 9j+-- .■.l + 2S=2(l-l + |-^ + l-l+...) = 21og.2. 12. The general term of 2, 3, 6, 11, 18,... is 7i=-2ra + 3; 7i^-2re + 3_ TO(n-l)-B+3 _ 1 1 3 •'■■"»" j^^ ~ [n ~ ]^r^ ~ |n-l + |ji^' q Q -I R 1 1 Thus "i = jl-^5 "2 = 11^-11 + 1' "s = |3-]2+i£= hence as in Art. 404, Ex. 1, 5=3 (e-l)-« + e = 3 (e-1). 1 + 1 „, 12-1 ,.|3 + 1 ^ |4-1 1 + 1 12-1 13 + 1 I 13. S=H-(l-l)x + ^i-a:i'-i=|-ic3 + i=j-x*-i ^ ■!/ at- ily B^ »C' = l + ^+j2_+^+J4+|5_+16_+- 15+1 / a;= x' a* x' x« \ -(="-2-+3-4+5-6+-) =e*-log(l+a;). 14. (1) Assume 16 + 25 + 35 + .. .=.4„m'+^in8 + ^n5+...+4„ then as in Art. 405, ()i + l)«=^ {(n + l)'-»'}+^i {(» + l)«-n6} + .... Equate coefficients of t^, re^,... [the various coefficients are given on p. 320]. Then l = 74o; ^0 = 7- 6=214„ + 6^; A^=\. 15 = 35^ + 15^ + 5^2; ^2 = ^. 160 SUMMATION OF SERIES. [CHAP. 15 = 21^0 + 15^1 + IOJ2 + 3^4 ; ^4= - 1 . 6 = 7^0 + 6-^1 + 5^2 + 3-^4+2^6; ^5=0- 1=Aq + Ai + A2+A^+Aq', Aq = y2» And by putting n = l, we have 1=A^ + Ai+A2 + Ai+Ae + A,; A,=0. (2) Assume l'+27 + 3' + ...+K'=^(,n8 + ^in* + ^2n<'+... + ^,n + ^8; then (n + l)'=^{(n + l)8-»8}+^^{(„+l)7_„7} + _..+j^. 7=28^ + 7^1; ^i=|. 21 = 56^ + 21^ + 6^2; ^2=^. 35 = 704„ + 35-11 + 15^3 -1-5^3 5 ^3=0. 35 = 5640+35^1+20^2 + 444; ^i=-^- 21 = 28^0 + 21^1 + 1542 + 644 + 3^5; 4g=0. 7 = 84o + 74i+642 + 444 + 24e; -^6=^- l=4o+4i + 42+44 + 45+47; 47=0. Puttingn = l, l=4o + 4i+42 + 44 + 4j + 48; 48 = 0. (m + l)8_ «(»i-l)(n-2) + 6 m(ra-l) + 7r. + l 15. "«+i--j^ ^^ 1 _6^_ _7 J. ~in-3 lji-2"'' Iw-l"*" Ik' 1 17 1 7 fl Thua«i = l; «2=ij + 7; '^=T2 + ]l + ^' "*~l3 "'"]2 ■'"'ii'''-^' hence as in Ex. 1, Art. 404, S= (1 + 7 + 6 + 1) e = 15e. XXIX.] SUMMATION OF SEEIES. 167 16, The required coefficient is equal to the coefifioient of a;""^ in 1 or {1-xf-cx (1- Bxpanding by the Binomial Theorem, the last expression becomes 1 , ex c^x^ c'x' (1-xY^ (1-x)* ' (1-a;)' ' (l-a;)s and we have to pick out from the expansions of [l-x)-\ (l-x)-\ (1-x)-', a-x)-8,... the terms involving s"-^, a"-', x""', x"— ',... respectively, and multiply them by 1, c, c^ c^.... 17. (1) This is the particular case of Ex. 3, Art. 404, in which « = 1, 6 = 2. (2) By putting a = l, 6=1 in Ex. 3, Art. 404, and proceeding as in that example, we have S = the coefSeient of x" in :; s l-x + x^ 1 + X = the coefficient of x" in -, 1 + x' = the coefficient of x"in (l + x)(l+x3)-i; and since n is a multiple of 3 the coefficient of x" is unity and is negative when n is odd, and positive when n is even. Hence S={- 1)". 18. K (x + l)" = X» + CiX''-l + C2X"-2 + C3X''-3 + ___. then (e>: + l)"=c»"^ + Cie"»-«"^ + C2c"^»^+..., {e« - 1)«= e"" - cje'"-"* + CaeC-^i + . . . ; .-. 2{e'« + C2e<»-«»' + C4e<''-«"^+...} = (c»^ + l)" + {«"^-l)'*. Equating coefficients of x', we have ?^ = the coefficient of x' in (e* + 1)» + (e^ - 1)» li / x^ x^ \^ = the coefficient of x' in I 2 + x + -^ "*" fi""*" •• ) ' that is, in n. 2"- (x+ J + |^) + "^^ . 2- (x + f)^ 168 SUMMATION OF SERIES. [CHAP. . 2S n.2-i n{n-l) n{n-l){n-2) ■• 6 ~ 6 "^ 2 '^ + lb 2n-3 = -g- {4n + 6n(n-l) + 7i(m-l)(»j-2)}; .-. S=n2(„^.3)2n-4. 19. (1) "«=i + „2 + „4 = 2(^l_„ + „2-i + „ + „2J; "'"^»~2\-^ l + »i + nV' (2) For the odd terms, m„ = — ; ;- = : and for the even ' " n{n + l) n n + 1 terms, i/„ = — — = =■ . Hence " n(n + l) n + 1 n and »-=(M)-G-5)-(i-i)-(i-i)--(i-^)^ S2™4.i= (i - a) + (2 - 3) + (3 - 4) + (i - 1) + (2;;rri - 2srr2) = thatis, «2™=3-2il>^°'^^=™+^ = ^-2Sr;25 " n + 1 _lfl 2 1 >, n(n + l)(n + 2) 2' ( x^ a? \ 2/a:2 a;' x* \ = i^-2-+3---j-5U"3 + 4--j . 1 (x^ X* x^ \ + ^(,3-4+^--) = log(l + a;)-^{x-log(l + x)} + i |-a; + |-%log(l + a:)l /, 2 IV, „ > 3 1 = (^ + S + ^)l°S(^ + *)-2-^- XXIX.] SUMMATION OF SERIES. 169 21. (e'^ + 1)" = c"* + Cie**^"* + e2c'"-2>"^ + CjeC'-'"' + . . . {e" - 1)"= e"»; - Cjc("-«^ + cje"'-^^ - Cje'"-"* + . . . .-. 2{Cie'»-«»+C3e»-3W + C5c(''-«"=+...} = (e"'+l)n-(e»:-l)". Equate ooefBoients of x^; then, if S denote the required series, 2(S we have —= the coefficient of s^in (e^ + l)"- (e'-l)"; 11 that i§, in f^ + x+'^X or in 2^-^n(x + ^\ + "^"~^^ 2''-=x2; .-. S=n.2»-2 + n(m-l)2»-'=n(ji + l)2"-». 22. (1) When » ia odd, - 2" _ 1 / 1 1 \ _ "»-(2'.-l){2"+i+.l) ~3 U»-l''"2»+i + iy ' and when n is even, «„= ^2„^i)^3„+i_i) = J (gS^ + 2^+^ : and 3S,„^,= (1 + 1) - Q + ^) + Q + 1) - (^ + ^) + ... + \^22»>+i-i "^ 2^+2:ri j ; 1 that is, 352ot=1- 2n+i + {-l)»+i' (2) The general term of the series 7, 17, 31, 49, 71,... is 7 + 10(7i-l) + 2(n-l)(«-2) or 2n^ + in-l; 2»i° + 4n + l ^ 1 /"l , _?_ 1 y n i 1\ /I 2 IN /I 2 IN .•.2S„=(j + 2 + 3J-(2"'5+i) + (3 + 4 + 5)-- = i+^+(-1)'-^^i+M)"-\T2- 170 SUMMATION OF SEMES. [CHAP. 23. Assume (l + ax){l + a^x){l + d''x)... = l+AjX + A„x' + J^^ + ...; change x into a^x ; then (l + a^x)(l + a^x){l + a7x)... = l + AiW'x + A2a*x^ + A3a^x^+...; .: {l + ax)[l + A^a.^x + Asa*x^ + A^a^x^+...)=:l + Aj^x + A^^ + A^^ + ...; .: a+Aj(i'=A^; hence Ai= _ g . A,aO+A,a*=A,; hence ^,=^^, = ^j-^^i-^. A,a^+A,aO=A,; hence ^3=^ = (i_„.)(i!l4)(i_^.) • 24. Here(l + x)^(l+|y(l+|y... Change x into =; then ('-I)'('*5)'('-S)"- =i+^ix+...+^r-ia;'""^+^ri'''+...; Now ^o=l> ^^^ Ji=suni of coefficients of x in the component factors [Art. 133] Putr=2, then^2=-(4 + l) = ^; 25. Let (1 + X)"=1 + C1X + C3X2 + C3X3^. . then (l+!a:)"=l + iCiS-C2a;«-iC3a;» + C4X< + iC5xS-,..; {l-tx)''=l-ic,x-C2x2+ic3xS + C4X*-icja:»-...; .-. 2ix (ci - C3X2 + Cja;* - . . .) = (1 + ii)" - (1 - ix)». 1072 3X5 ' XXIX.] SUMMATION OF SERIES. 171 Put a!''=3, so that a:=^S, and let Sj denote the value of the first series; also as usual let u, ur^ be the imaginary cube roots of unity ; so that w^ ^^ . ar= — . 2 ' 2 We have 2 V3 • Si = (1 + s/^)» - (1 - s/^)" = ( - 2a>=)'' - ( - 2ai)» = (2)" - (2)" = 0, ■when n is a multiple of 6, for then ( - w)''= 1, ( - &>'')"= 1. Put x^= -, and let S^ denote the value of the second series ; then = ( , I - ( " 1 =0, if mis a multiple of 6. \^-3j \J-3J 26. As in Example 3, Art. 401, we may shew that the given series is equal to the coefficient of x" in ■: — = x, where b=p + q, a=pq. In this X — ox T UiX case 1 _ 1 ^ [ P i_l . 1 — bx + ax^ l — {p + q)x+pqx- p — q\l—px 1 — qxj' whence the result at once follows. 27. P,= |i> X the coefficient of x"-'-! in (1 - x)-i-7)(«-6)(n-5) .g ^j^g coefficient of x--^ in j (1 - x)-"; 172 SUMMATION OF SERIES. [CHAP. and so on. Henoe S=the coefficient of a;" in a;=(l-x)-i-^a;*(l-x)-2+ix6{l-a!)-3-jx8(l-x)-»+... = the coefficient of x" in log{l + x2(l-x)-i}. But l + x2 l_a:)-i = l+- = -_ = - ; ' 1-x 1-x 1-x^ /. S=coefficient of x" in log(I + x')-log(l-x'^). If m=6r, the coefficient of x" is -^r- from the first series, and — from 2r 3r the second, /. S= = — . 'or n If re=6r + 3, the coefficient of x" is ;; ^ from the first series, and zero 2r + l 3 from the second; thus S= - . 29. X x' x' x' l-x« ' 1-x" l-x» "*■••• l-x» =x +x' +x5 +X'' +x' +x" + .., -X3-X9 -X»5-X21-X27-X33-.., + x5 + X« + x2S + x35 + a« + X=5 + ... -X7-X21-X35-X<9-X63-X''-.... By adding the vertical columns, we obtain X x^ s? x' 1 + X2^1 + X6^1 + Xl»^l + X"^ EXAMPLES. XXX. a. Pages 348, 349. 1. 3675 = 3 . 5= . 7=; thus the multiplier is 3. 4374 = 2 . 3'; thus the multiplier is 2 . 3 or 6. 18375=3 . 53 . 7-; thus the multiplier is 3 . 5 or 15. 74088=23 . 33 . 73 . thus the multiplier is 2 . 3 . 7 or 42. 2. 7623 = 3^ 7 . 112 ; thus the multipUer is 3 . 7M1 or 1617. 109350=2 . 52 . 3'; thus the multiplier is 2- . 32 . 5 or 180. 539539 = 7M1M3; thus the multiplier is 11 . 13^ or 1859. 3. If x-^ is even, then x-y + ly, or x+j/ is also even; hence x-y and x + 2/ are both divisible by 2, and therefore their product is divisible by 4. 4. Let n be the number ; then the difference =n^ -n=n{n-\)\ and one of the numbers n, ra - 1 must be even ; hence the result. XXX.] THEORY OF NUMBERS. 173 5. ix^ + 7xy -2y^ = {4x-y) {x + 2y); since ix-y is a multiple of 3, it follows that ix-y-S{x-y) or x + 2y is also a multiple of 3; thus the expression is divisible by 3 x 3 or 9. 6. 8064=2' . 3 . 72; henoe by Art. 412, the number of divisors = (7 + 1) (1 + 1) (2 + 1) = 48. 7. 7056 = 2* . 82 . 72 ; henoe by Art. 413, the number of ways =- {5. 3 .3 + l} = 23. 8. 2*"-l = (2^)''-l"=16"-l", and is divisible by 16-1, or 15. 9. M(n + l)(n + 5) = n(n+l)(n^ + 6) = (7i-l)n(n + l) + 6«(n + l); and each of the terms of this last expression is divisible by 6. 10. The difEerence between a number n and its cube = n' _ n = 71 (n - 1 ) (re + 1) = (n - 1) m (n + 1) , and this being the product of three consecutive integers is divisible by 6 ; hence re* and re when divided by 6 must leave the same remainder. 11. m(re2 + 20) = re(re2-4 + 24) = re(re-2)(re + 2) + 24n. Now (re - 2) « (re + 2) is the product of three consecutive even integers and therefore must be divisible by 2 . 4 . 6 or 48 ; also 24n is divisible by 48 ; hence the result. 12. n (n" - 1) (3re+2)=re (n + 1) (re- 1) (re + 2 + 2re) = (re-l)re(re + l)(re + 2)+2n(re-l)re(n + l). This last expression consists of two parts, the first of which is divisible by |4 or 24. [Art. 418.] The second part is divisible by 3 ; it is also divisible by 8, for if re is even, 2re2 is divisible by 8, and if n is odd 2 (re - 1) (re + 1) is divisible by 8 ; thus the second part is also divisible by 24; henoe the whole expression is divisible by 24. 13. re6_5„3+4„=„(„2_4)(„3_x) = (n-2)(re-l)re(re + l)(re + 2), which being the product of five consecutive integers is divisible by 15, or 120. 14. 3'>» + 7 = (32)"-l + 8 = 9'»-l"+8; now 9"-l» is divisible by 9-1 or 8 ; hence the result. 15. Since re is prime to 3, re^ _ 1 is divisible by 3 [Art. 421]. Also since mS_ l = (re- 1) (re + 1), it is the product of two consecutive even integers, since re is prime. Thus the expression is divisible by 2 . 4 . 3 or 24. 16. re' - re is divisible by 5 [Art. 422]. Again re'-re=re(re*-l) = »(re-l) (re + 1) (re^+l); and this expression is divisible by 13 or 6. Thus re« - « is divisible by 5 . 6 or 30. 174 THEORY OP NUMBERS. [CHAP. Again if n is odd, the expression n{n-V){n+\) (n?+l) is divisible by 240 ; for the product of » - 1 and ?i + 1 is divisible by 2 . 4 or 8 ; one of the first three factors is divisible by 3 ; and n^ + 1 is even, since n is odd. As in the first part of the question the expression is divisible by 5; thus it ia divisible by 2 . 4 . 3 . 2 . 5 or 240. 17. Let m and n be any two prime numbers greater than 6. Then m' -n^=(m^ — l) — (li? — \) ; and each part of this expression is divisible by 3. [Art. 421.] Also each part is the product of two consecutive even numbers, and therefore divisible by 8. Thus m^ - n^ is divisible by 24. 18. If possible suppose N^=--3n-l, then N^ + l = 3n, a multiple of 3. But this is impossible for N^ + l = {N^-l)+2, and by Fermat's Theorem ^^ - 1 is divisible by 3 when N is prime to 3 ; thus ^^ + 1 exceeds a multiple of 3 by 2; and therefore N" is of the form Sn + 1. It N is not prime to 3, it is clear that N^ must be of the form 3n. 19. Every number x is one of the forms 3g, 3g±l. If x = 3q, then x^=27q^, and is of the form 9n. If a; = 3g ± 1 , then x^ = 27s= ± 273^ + 9g ± 1, and is of the form 9n ± I. 20. -Z^ is either equal to In, or else is prime to 7 ; in the latter case N^-1 is a multiple of 7, and therefore either J^' - 1 or ^7^ + 1 is a multiple of 7. Thus every cube number is of the form 7n or 7n ± 1. Also 7™ - 1 = 7 {re - 1) + 6 ; therefore if N^ is divided by 7, the remainder is 0, 1, or 6. 21. Let the number be N^; then if J?' is a multiple of 7, N=7n; if N is prime to 7, N''-l=iln, oi N^='ln+1. X ix-{-y) 22. Let be the triangular number. Then this is a multiple of 3 if either x or x + 1 is divisible by 3. If neither x nor a:+ 1 is divisible by 3, 1 9 a; must be of the form 3re + l; in this case ^x{x + l) = -n(n + l) + l and is therefore of the form 3r + 1. Thus the form 3n - 1 is inadmissible. 23. Let r, s represent any two of the numbers 1, 2, 3,...ra; also suppose that r^-s^ is divisible by 2re + l. Now 2ra + l is prime; hence either r + s 01 r-s must be divisible by 2re + 1 ; but r and s are each less than re, so that r + s and r-s are each less than 2re + 1 ; hence r^ - s^ cannot be divisible by 2)1+1, that is r^ and s^ cannot leave the same remainder when divided by 2re + l. 24. If a is odd, then a" is odd; hence a^ + a and a" -a are both even. If a is even, then a" is even; hence a'^+a and a' -a are both even. 25. (2a; + 1)«« = (4a;2 + ix + 1)" = {4a; (a; + 1) + 1}»= (8m + 1)", because a;(a; + l) is even; but (8m + l)"=8r + l; hence the result. 26. I'rom Fermat's Theorem, by putting p = 13, N^''-l = M[13) = 13n, when N is prime to 13; thus N^^=13n + 1. If Jf is a multiple of 13 then evidently ^12 =13re. XXX.] THEORY OF NUMBERS. 175 27. K -W is not prime to 17, then N« = lln. If N is prime to 17, then by Fermat's Theorem W'- l = Jlf (17); that is, (J\r8+1)(2^'-1)=M(17); hence ir8±i = 17„. or N^=nnJ^l. 28. We have m* - 1 = (n + 1) (n - 1) (»" + 1) ; and (n + 1) (re - 1) is divisible by 8, being the product of two consecutive even numbers. By Fermat's Theorem n'' - 1 is divisible by 3, and n* - 1 by 5 ; also n' + l is even. Hence n* - 1 is divisible by 8 . 3 . 5 . 2 or 240. 29. m° - 1 is divisible by n* - 1 and therefore by 8, when re is prime. Also re^ - 1 is divisible by 3, from Fermat's Theorem ; and re° - 1 is divi- sible by 7, except when n = 7 ; thus re' - 1 is divisible by 8 . 3 . 7 or 168. 30. n^-l = (re" + 1) (re* + 1) («» - 1) , and each of these 3 factors is even ; and of the last two factors one is divisible by 2 and the other by 4 ; thus n^ - 1 is divisible by 2 . 2 . 4 or 16. Again re' - 1 is divisible by 3, n" - 1 is divisible by 19, and n^ - 1 is divi- sible by 37. [Art. 421.] Thus »3«- 1 is divisible by 16. 3 . 19, 37 or 33744. 31. Since x is odd, x^p-1 or (x" + 1) (x" - 1) is divisible by 8. Again by Fermat's Theorem x^-l is divisible by p + 1, and x* - 1 is divisible by 2p + 1 ; whence the result follows at once. 32. By Fermat's Theorem x^-' -l=M{p); hence x"-' =l + kp; .: (xJ'-i)P'~'= (1 + kp)p''~'- = 1 + kpM {p'-^). .: x!''"-J'''"^=l + JlI(p'-); which proves the proposition. 33. Here both a and b are prime to m; hence by Fermat's Theorem, (jin-i_l and 6"'~i-l are both multiples of m; hence their difference a™~' - 6"""^ must also be a multiple of m. Since a and 6 are both less than m, their difference a- 6 is less than m and therefore prime to m; hence (a"*"!- 6'"~^)-i-(a- 6) must be a multiple of m; that is, a"^' + a^-^b + a"*-^62 ^ ^ jm-a is a multiple of m. EXAMPLES. XXX. b. Pages 356—358. 1. Let/(re) = 10'' + 3.4''+2-|-5; then / (re + 1) = IC+i + 3 . 4»+3 + 5 ; .■./(re + l)-/(re) = 10" (10-1) 4-3. 4''+2(4-l) = 9. 10"-l-9. 4''+2=il/ (9). And /(l) = 10 + 8.43-|-5 = 207=M(9). 2. Let /(n) = 2.7» + 3.5'»-5; then/(n-|-l) = 2. 7''+i + 3. 5"«-5; .•./(n + l)-/(n) = 2.7".6 + 3.5".4=lf(24), for 7'' + o"iseven; and /(l) = 2.7 + 3.5-5=24. 176 THEORY OF NUMBERS. [CHAP. 3. Thig Trill follow if we shew that 4.6" + 5"+! - 9 is divisible by 20. Let /(k) = 4.6''+5»+1-9; then /(?i + l) = 4. 6''+i + 5»+2-0; .-. /(ji + 1)-/(b) = 4 . 6". 5 + 5»(52-5)=M(20) ; and /(l)=4.6 + 52-9 = 40=Jlf(20). 4. 8 . 7" + 4''+2=8. T^ + 2^'^*=8 (7'> + 2^"+i). Let f{n) = 7" + 22"+' ; then / (n + 1) = 7"+^ + 2^"+" ; /(n + l)-/(n) = 7''.6 + 2''+i.3=il/(3); and /(l) = 7 + 23=16=Jlf(3); thus 7» + 22''+i is divisible by 3; but 7" is odd and 22»+i is even, hence the quotient must be odd, and therefore of the form 2r - 1. Thus 8.7" + 4»+2 = 24 (2r - 1). 5. By Wilson's Theorem, l+\p-l = M (p); ■■■ l + {p-l){p-2)\p-3 = M(p); .: l + {p^-3p + 2) lp-3 = M(p); .: l + M{p) + 2 \p-3 = M{p); whence the result follows at onoe. 6. a*+i-a=a{o*-l), and is therefore divisible by a (a^-1) or a^-a; hence by Art. 422 the given expression is divisible by 5. Similarly it is divisible by a (a" - 1) or a' - a and therefore by 3. If a is even, the expression is clearly divisible by 2, and if a is odd, a*' - 1 is even and the expression is again divisible by 2. Thus the given expression is divisible by 2 . 3 . 5 or 30. 7. The highest power is the sum of the integral parts of the expressions 2'--l 2''-l 2''-l 2'-l 2~ ' 2^ ' 23 ' ■■■ 2''-' ' and is therefore equal to (gr-i _ 1) + (2'--= _ 1) + (2'-3 - 1) + ... + (2 - 1) =1^ - (r - 1) = 2'- - r - 1. 8. Let / (») = 3''"+= + 52"+i ; then / (n + 1) = 34"+6 + S^^+s ; .-. /(ii + l)-25/(n) = 3^"+2(81-25)=lf(56)=21f(14); also / (1) = 3« + 5' = 729 + 125 = 854 = M (14). 9. Let /(») = 32»+5 + 160»i2-56»-243; then /(« + l) = 32»+7 + 160(»+l)i'-56(n + l)-243; .-. 9/ (»)-/(» + 1) = 160 (8»2 _ 2n - 1) - 56 (8k - 1) - 1944 = 1280»2 _ 768ji - 2048 = 256 (o)i2 - 3» - 8) = 256(5n-8)(» + l); and it is easy to shew that (5ra - 8) (ra + 1) is divisible by 2 ; .-. 9/(m)-/(» + l)=Jlf(512). Also /(l) = 3' + lG0-56-243 = 2048=M(512). XXX.] THEOET OF NUMBERS. 177 10. Let (l+x+a?+afl+x*)'^^=l + CjX+c^' + CgX^ + .., then {l-x+x'-a^+x*)"-''^=l-CiX+c^^-c^+..„ Let S=e^ + Cg + Cg + ..,; by subtracting and putting a=l, we have 2S=5"~i-l = M(n), by Fermat's Theorem ; hence S is divisible by n, since n is prime and greater than 2. 11. n' - 1 = M (7), by Fermat's Theorem. Again n' - 1 is divisible by n^ - 1 and therefore by 8. Since n must be one of the forms 3q + l or 3g - 1, it is easy to see that m^ - 1 is divisible by 9 ; hence the given expression is divisible by 7 . 8 . 9 or 504. 12. n'+3n'^ + 7n^-ll = {n^-l){n* + in' + U)=M(S) . {n* + in^ + n). And n* + 4»2 + ll = (ni'-l)(n2-ll) + 16n2=M(16), for n'-U is even. Thus the given expression is divisible by 8 x 16 or 128. 13. Let the coefficients be denoted by Cj, o'l, Cj, ... c, Then c _ (P-l)(P-2)(P-3)-(i'-'-) M{p) + {-ir\r_M{p) \L \L + (-!)' M (p) Now c, is a positive mteger; hence — p- must be a positive integer, and since ^ is a prime number, it must be a multiple of p ; therefore c^=Jf(i)) + (-ir. 14. Li Art. 426, Cor., it is proved that if the p terms of a series in A. P. are divided by^ the remainders will be 0, 1, 2, 3, ... ,jp-l. Hence disregarding the order of the terms, the series may be represented by op, bp + 1, cp + 2, dp + S, ..., kp + (p-l); a, b, c, d, ... fc being the various quotients. With the exception of the first term all the terms of the series are prime to p; hence by Fermat's Theorem, their {p - 1)"> powers are all of the form M (p) + 1, whilst that of the first term is of the form M (p). Thus the sum of the (i) -l)"" powers=Jlf (p) +p-l = M{p) - 1. 15. (a" - 1) - (612 -i)=M (13), by Fermat's Theorem. Similarly, (aS-l) - (t'=-l) = M(7); hence a^^-b^^ is divisible both by IB and 7, and therefore by 91. 16. By Wilson's Theorem, 1 + | j)-l =M{p) .: l + (p-X)(p-'2) {p-2r-l)\ p-2r =M{jp) .: l + {M(p)- \2r-l } \p-2r =M{p) .: 1 + M {p) - \ p-2r \ 2r-l = il (p) whence the result follows. H. A. K. 12 178 THEOBT OF NUMBERS. [CHAP. 17. Since (n-2) (m-l) «(» + !) (*s + 2) is divisible by [5 or 120; and n-1 and n+X are both prime and greater than 5, .•. m(n-2) (n+2) la divisible by 120, that is, m(n»-4) is divisible by 120. Again {n - 1) n (n + 1) is divisible by 6, and therefore n is divisible by 6 since n-1 and m + 1 are prime and greater than 5. .-. n" {rfi - 4) is divisible by 720 ; also 20?i- is divisible by 720 ; .-. n^ (rfi - 4) + 20m2, or n^ («=+ 16) is divisible by 720. Lastly n= 6s, and one of the three nmnbers n + 2, n, n - 2 is divisible by 5. (1) If 7i = 6s = 5r-2; s + ^-~=r; s = 5t-2; .: n=30t-12. (2) If n=6s = 5r; n=30t. (3) If n = 6s=5r + 2; s + ^-^=r; s=5t + 2; .: n=30t + 12. 18. The highest power required is equal to the sum of the integral ji'-l n'-l n'-l n'-l „ , ,„ parts of —^, -^, ^....^icr; [Art. 416] that is, = («'■-' - 1) + (n'-" - 1) + (n'-' - 1) + ... + (n - 1) n'-n , ,, n''-nr+r-l =^rri-('-i)=^ — 19. We have c^ - a= kp, so that a—c' - kp ; .-. a^^'"'-l = {c'-kp) 2 -l = cP-i-M{p)-l, since y - 1 is an even integer. Also a is prime to p, so that c must be prime to p ; hence c'~^ -1=31 {p), by Fermat's Theorem, and the result at once follows. 20. The congruence 98x-l = (mod. 139) means that 98a; -1 is divisible by 139 ; ily is the quotient, then 98a: - 1 = 139i/, or 98a;- 1391/= 1. 139 If — — is converted into a continued fraction the convergent just preced- ing the fraction is j^ . Hence the general solution of the equation is a;=61 + 139t, ^ = 43 + 98t. 21. The numbers less than N and not prime to it are given by >S--S — + 24--... [See Art. 432.] a ab aoc "■ ■" Let us first find the sum of the squares of all numbers less than N and not prime to it. XXX.] THEORY OF NUMBERS. 179 These are given by the snm of a'>+(2o)2 + (3a)s+...+ (--aY + 6=^ + (26)!! + (35)2+ ... + Ty . ftV - (a5)2 - (2a6)2 - [Sab)" - ... - f^ . abX - (ftc)" - (26c)a - (36c)a - ... - (|^ . ScV + (o6c)2 + (2a6c)2+ (3a6c)H ... + (^ . abcY Now oa+(2a)2+ (30)2+. .. + (-. a)" ,', the sum of the squares of all numbers less than N and not prime to it is Jf3 + - N^ il 1 1 11 1 1 -5- -^- + T+-+... r ... + ^-+...V 3 [a o c ab ac abc j N^ I m(m-l) m(m-l)(m-2) 1 ^ r — 172"+ 17273 -f N +-^ {a+b + c + ...-ab-ac- ...+abe + ...}, where m is the number of prime factors in N. Thus the coefficient of -^ = 1-(1-1)'»=1. .•. the sum of the squares of aU numbers less than N and prime to it is obtained by subtracting the above expression from -^{N+1)(2N+1), 01-^+-^+-^. .'. the sum required is 3 (, 1 1 1 11 1 1 [abc ab ae abc f N + -g^{l-a-b-c~ ...+ab + ac + ...-abc-...} 12—2 180 THEORY OF NUMBERS. [CHAP. To find the sum of the cubes we may conveniently use the following method. As in Art. 431 let the integers less than N and prime to it be denoted by li P, i, r,..,N-r, N-q, N-p, N-1. If X stands for any one of these integers, then Sa;'=S(jy'-x)'; for each of these expressions denotes the sum of the same series of terms, only the order in one is the reverse of that in the other. ' Hence 2x^ = J:N^-S'2N^x + 3-ZNx^-7:x^; that is, 21,x^=N^ (d,) + {d^ + (f> ((Jj) + ...=N; hence the above series =y + 3y^ + 5y'^ + 7y'' + 9y^+... =y(l + y''){l + 2y^ + Sy*+iy^ + 5y« + ...) =y(l + y'){l-y^'\ Writing ix for y, the required theorem at once follows. EXAMPLES. XXXI. a. Pages 367—369. 1. Assume P„=anI>„_i + 6„P„-2. gn=On9»-l+*n?n-2; then, as in Art. 438, the (n + l)"" convergent ("n-f^^Pn-l-KP^-i i'n-r^i'n-1 „ 7, -b r, /'„ _^Ao -Jo o -*a+ia «n+l9B-6«+l?n-l' I "n T I ?n-l "b ?n-2 «n T Sb-1 \ «n+l/ »n+l Then by induction the required result follows. 2. We have (^-^^j = 4x^ =^ + ^^' 4.r3 _ a; 4a; + l ^^ 1_ 4x + l~ 4x + l' a; x' •• V 2x / X- 4+ X 3. (1) Since Ja2 + 6=a + (^/a2+6-a)=o+-p= ; * ' V« +*+« and Ja^ + b+a=2a + ya'' + b-a) = 2a+ .^^ ; V" +" + <* ... Va^ + 6=a + 2— 2^ 184 CONTINUED FRACTIONS. [CHAP, , , b (2) Again,- Jd'-b=a-{a- ^a?-b)=a- --^=^\ , , „ 6 a+ Ja''-b=2a-(a-ija^~b)=2a r=F=^5 a+isja^-b l-T-T * * ^ 2a- 2a- 4. As in Example 1, Pn='^nPn-i - KPv.-^'' •■■ i'»-i'n-l=(«n-l)i'n-l-*nJ'7>-2- Nowa„-l is at least as great as 6„; therefore p^-Vn-i is at least as great as 6„(p„-i-f„-2); therefore l)„>p„-i if i'„-i>Pn-2; a^d so on. But i)2 is clearly greater than Pi; hence jp„>2)„_i. Similarly 3„> 3»-i. 1 1 2 5. By definition, 1 = ; **n ^n-2 ^n-\ ,. £«=_i = 2-^^; that is, -^ = _i_; «n "^n-S "»-l 2 "n-1 "n-2 similarly, '^^^ = — i — ; ; and finally ^ = — - . «n-2 2-^2=? ^ 2- — a«-3 «i 6. Denote the continued fractions hy a; and y ; then x-a=- = : whence i^-a'=l, or x=Ja'+l. 2a + x-a x + a Again, 2'-"=-2^T^r^ = ^ = .-. y^-a^=-l, or y=Jd'-l; ■^\ience.x^ + y^=a? + \ + a?-l='ia^. Finally, x!/ = 7«*^=«''-(a''-N/«*-l) = «^ — ; 7=5 a'+ Va*-1 'l'"^ ^2' = "'-2H^ 2^ 7. The series P=^i+^2a;+P3a;''+;)4a;' + , and Q=q^ + q^x + q3X^ + q^x^ + , are both recurring series in which the scale of relation is 1 - ax - bx\ XX.XI.] CONTINUED FRACTIONS. 185 "~ 1-ax-bx^ 1-ax-bx'^' Q^ gl + (g2-'»gl)'g . 0,+ bx xu = - 1-ax- bx^ \-ax- bx^ ' ax + bx^ 1 1-ax-bx^ 1-ax- bx^ Thus ^^1 = j,,=the coefficient of x" in = ^— r-,- 1 - aa; - bx^ Again 6 2^1 - op^i =J5^ - o^)^! = 5p„ = J'j^^j. 8. Proceeding as in Example 7, we see that p^. = coefficient of y^^ in 2 , and 53.= coefficient of y" in : l-ay-by-^' ^* " l-ay-by^' If a, |3 are the roots of i''- aft- 6 = 0, then a+p=a, ap= -b, and 1-ay-bxf l-(a + /3)2/ + a^i/2 (\-ay)(\-^y) a-^\l-ay l-§y)- 9. Let x=a + 5-- b+ c+ d+ a + ... ' 1111 b+ c+ d+ X „, .a 06 + 1 abc + c + a abcd+cd+ad + ab + 1 The convergents are t . — ; — • — 7 ; — > ^-^ — ; — ; ; " 16 6c + l ' 6cd + d + 6 ' _(abcd+cd + ad + ab + l) x + abc + c + a _ ~ (6cd + d + 6)i + 6c + l ' /. (6cd + 6 + (J) rc^ - (abed + ab + ad-bc + cd)x- (abc + c + a) = 0. If y=-d + ——- —r— -— ——J—..., by writing -d, -c, -6, -a for — c+ —6+ —0+ — 0+ a, 6, c, cZ respectively, we have ( - a 6c - c - a) y" - {a6cd + cci + O(i-Jc + a5)y-(-6cd-5-d)=0; or (abc + c + a)y'' + (abed + ab + ad-bc + cd)y ~(bcd + b + d) = Q. Now y is the negative root of this equation ; by putting y= — we have (6cd + 6 + d) z^- (abcd + ab + ad-bc + cd)z-(abc + e + a) = Q, and therefore z = x; that is 2/ = — , or a;?^ = - 1. 186 CONTINUED FEACTIONS. [CHAP. (n' - 1)^ 10. Here v— ; — Vr5 is the (»+!)"' component. or M^i - n* M„ = (n + 1)2 {«„ - (m - 1)2 m„_i f , similarly «„ - (m - 1)* «„_i = n* {w„_i - (re - 2)2 «„_„} ; M3-22«2 = 32(Mj-«i). Hence, by multiplication we obtain Now Pi=l, gi=l; j)2=5, 52=1. Thus 3»+i-«''?»=0, or g„4.i=n2j„. Hence 2n+i=«'' («- 1)^ (n-2)2 ... 12 = (|n)». Again j)^, - 7i2^„=32 .4?... (n + lf .4 = (|« + 1)'. • ^"+1 ^»_-f-, + i\2. . and -^2_ _^_o2 •■(|n)» (|re-l)'~^"+^^ ' ^'"^ (\Vf T-^' Hence, by addition gi-l = 22 + 32+ + (re + l)2; •■• (^g. = l^ + 2' + 32+... + (re+l)n andg = lj _. .Pn+i_x2|g2| I („| ^)2_ (" + l)(re + 2)(2n + 3) ^ 11. Here «„=(2n + l) «„_i-(»2-1)m„_2; or «„ - ««„_! = (« + !) {a„_i - (n - 1) «„_„} ; «3-3U2 = 4(«2-2«i). Hence, by multiplication we obtain u„-n«„_i= (re+ 1) n ... 4 (1*2 - 2«i). Now i)i=2, gi=l; J)2=10, g2=2; .-. i)„ - n2)„_i = |re + l , 2„ - ng„_i = 0. Hence j„= 7ig„_i = n (n - 1) 3„_2 = . . . = |n. XXXI.] CONTINUED FRACTIONS. 187 •whence, by addition ^-2 = 3 + 4 + . ..(re + l); In 1.-U 4. • Pn n(n+S) , ,, , p. nte + S) that IS, -fs = ' ^ ' ■ and therefore — = ' „ ' . [n_ 2 ' Sn 2 12. Here «„+! = (m + 2) «„ - (n + 2) m„_i ; M,,.j n + 1 n m + 2 " "^ 71+1 " n " ^ '■ n + 2 n+1 \n + l n J ' Hence, by multiplication 71+2 71+1 L^S 2J' Now i'i=2, Si = 2; i>2=6, 53=8; •■•ff2-^=°= sothat|^ = ^=?^-... = | = l. ^B- 1^2- 7.^-1= l!i= '-^t-t=li= by addition, |^= 1 + [1^+ |2 + [3_+ ... + [n ; hence we have the required result. 13. Here «n+i=(« + 2)M„-nu„_i; «3 — Sttj = U2 — 2tii. Hence, by multiplication ".tfi - (n + 1) «„=«a - 2«i. Now ^1 = 1. 9i=l; ^2=3. S2 = 2; ••• 9»+i-("+l)9n=0; ••• gn+l=(»+l)gn=(" + l)»gn-l=- = |" + ^ - 188 CONTINUED FRACTIONS. Again i'»+i-(»+l)i'n=l; . Pii+l Pn 1 [chap. ■■ \n + l |7i_ |»+1' • ^"^^ j2_ (1 - |2_' .^=^' = l + T^+ii+... + r^; so that ?= = c-l. 14. Here «„ = nM„_i + (2n + 2) «„_;, ; ■■• «» - (« + 2) «„_1 = - 2 {«„_! - (71 + 1) M„_2} ; «3-5«2=-2(K2-4«i). Hence, by multiplication »„-(n + 2)«„_i=(-l)n-2 2''-2(«3-4ui); also Pi = 4, ?i=l; j)2=8, g2=8; /. i)n-(" + 2)i'„-i = (-l)''-i2"+i, ?„-(«+2)g„_i = (-l)»-22»; |« + 2 In + i (7! + 2 ' |n + 2 In + l gn-i _ (-l)"-^2" . n+2 ' J>2 i>i^ (-l)23 ?2_3j_2« (£ [3 [4 • [4-(3_-|4; Hence, by addition ^=|_|+|_._ and -1" -ij.?!_?!^?^ |w + 2 [3_^ |£ |5_"^ [6_" ^_l/'2'_2i 25_ \_l/'4 2* 25 „-2Vl3 |4_+|5 •■■; ■4Vl3_"*'[4-|5 Now ~ 1^ ]3+J4_- ■■?„ 2\, 1-2+--.- ■)*i« -—S.J) = |(l-«-=)-l(l + _2(l-e-'i)_2(e''-l) 1 + e- «2+l XXXI.] CONTINUED FRACTIONS. 15. «„=nM„_j + 3(7i + 2)M„_3; «„-(« + 3) M„_i = - 3 {u„_i - (re + 2) «„_„] 189 Uj - 61*2= - 3 (uj - 5ui). Hence, by multiplication «„-(« + 3) M„_i = ( - 3)''-2 («2 - 5ui). Now j)i=9, gi=l; y2=18, 32=14; ••• i'n-{™ + 3)i'n-i = (-l)"-^3''+i, s„-(n + 3) g„_l=(-l)»-=3^ Pn ft.-! _(-l)"-'3"+i |w + 3 |« + 2 |n + 3 _3n_ g.-l^ (-l)"-^ 3". |n + 3 |m + 2 |« + 3 ' p^ Pi_ (-1)3« £2_?i_3_\ ■whence, by addition p^ 32 33 , 34 i+z li l^ LI 1 /3< _ 3» 36 _ \ and Now Sn 1 32 33 3* [n+3 ^ ^ Li LL 32 33 3* !*_ _J_/'3f_3^ 3^_ \ 7~"'~27V15 16"'' 17 --)' 34 35 36 ■3„-9^' +2). 27(^1^ ^11 ; 5 6 (e-3 + 2) _ 6(2e3+1) 5 - 2e-3 -g' -g- -=s-- 16. Here where Pn=9'n-i. and 2„=g„-i+i'n-i = !?M-i + 9n-2' Hence gi + 32^ + g3a;^ + 24a;3+ ...is a recurring series in which the scale of relation is 1 - x - x''. Also 3i = *> q2=a + 'b. 190 CONTINUED FRACTIONS. [CHAP. Let . _ _o = i-— - + i l-x-x2 l-ax l-/3x' then g„=4o''-'+-B/3"-i; p^=q^_i=Aa'^^ + B^-^; •'• 8„~.4o''-i+JB|3»-i" Now a and j3 are to be found from the equations o+^=l, aj3= -1; let a be the greater of the quantities, then so that o > 1 and /3 < 1 ; hence the limit when »i is infinite of a" is » and of /3»isO; . j),_ ^a— ' 1_ 2 _ «y5-l ■■ g„~.4o''-i~o~l+,v/5~ 2 * 17. We have «„=(r+l)«„_i-™„; that is, M„-(r + l)u„_i+ru„=0. Thus the series Uj+u^ + u^x^+ ... is a recurring series, whose scale of relation is 1 - (r + 1) x+rx^, and whose generating function is Hi + {»2-(r + l)«i}x l~x — x^ Now Pi=r, qi = r + l, p^=r[r+l), q^=r'+r+l, _ r / r 1 \ ~r-l\l-«i!~l-a;y' •■•p»=fri('""-i)- r+l-rx Similarly gi + g2a; + g3x'' + g4a^ + ...= l-(r+l)a;+j-xi' r-lVl-rx l-icy' Thus we have |; = ^1- XXXI. J CONTINUED FRACTIONS. 191 18. We have M„=(o„+l)«„_i-a„«„_2; that is, «n-«n-l = anK-I-"n-2); «3-tt2 = a8(lt2-Mi). Hence, by multiplication Now Pi=ai, 2i = ai + l, j>2=ai(aj+l), 32=0102+ Hi+l; Pi= 2i = l + ai. Hence, by addition i>„= aj + Oia2 + aiaja3+... +010203... a„; g'„= 1 + «! + O1O2 + 010^03+ . . . + OiOjOj. . .a„. .-. 'i-+p„=qn; and p„, q„ are both infinite in the limit; hence the con- tinued fraction tends to the limit 1. .19. The convergents to ■^—- ;r-— r-— vt— ... are 1+ z+ 1+ J + 1 2 3 ^ 11 30 41. 1' 3' i' 11' 15' 41' Si' and the convei gents to 1 - j — v — j — ... are 1 3 11 41 153 I' 4' 15' 56' 209* •" Let — , — , —,...; — . -. — ,... denote the two sets of convergents; 2i 22 93 h h h ,..,,, then Pi=r^, P3=^a, Ps='''a, i'7=''4,-.. ; aud similarly for q and s. Now i'2n-l=Psn-2+i'2n-3' i'2n-2 = ^i'sn-S +i'2n-4 ' Pm-S= Pin-i+ Pin-l'< whence P^-i-iPin-3+Piin-6=^> but »-„-47-„_i+r„_2=0; thus P9=^P7-P6 = ^i-'^3='rS' Pii=iP9-P7=K-U=''ai hence generally i)a,_i=r„. SimUarly ?an-i=»n- 192 CONTINUED FRACTIONS. ' [CHAP. 20. We have ps„ = Psn-i-Psn-i' Psn-l = 2^8n-2 ~ Psn-S > Psn-2=^Psn-3-Psn-i' P3n-3= P»n-4-P3n-6> i'3»-4= 2y3n-6 -Psn-e- From the first three equations, p^n^ipan-a- P^n-ii from the last two equations 2p3n-3=P3n-4- Psn-s- ^7 combining these results we have Psn^ ^Pan-3 ~ P3ii-i 1 so that the scaleofreIationisl-2a; + a;^. Now 3)3=1, 33=4, 3)3 = 2, 58=7; Similarly ffs + Ss^ + 39^" + Ssn^""^ + . • • = x-%x + x^ ' •"■ i'3n="i ^■'^^ g'3„=4n-(ra-l) = 3m + l. 21. This may be proved db initio, or it may be deduced from the Example in Art. 444 as follows. 12345 11 3 2.4 5 1+ 2+ 8+ 4+ 5+ - 1+1+ 2.8+ 4+ 5+ ' 1 1 1 2.4 3.5 1+ 1+ 2+ 3.4+ 5 + 1 1 1 2 3.5 4.6 1+ 1+ 2+ 8+ 4.0+ 6+ ■■■ _J_ J 1 2_ 3_ "1+ 1+ 2+ 3+ 4+ - [Compare Art. 448.] Thu3 e-l^l+^^j:^.... If X denotes the value of the given expression, we have e-l=l + = ; whencea;= — -. 1+x c-2 1 ., 1 2 ' 3 Now ^<2and>— gorg. /, — 7,<7:and >~; that is, 8<3e and 30>llc, X.X.XI.] CONTINUED FRACTIONS. 193 1. Put EXAMPLES, XXXI, b. Pages 371, 372. 11 1 «r «r+l u^ + ^r' then {u,^.l - «,) (u^ + x^) = «,M^i ; u ^ 111 so that x-= , and therefore = i, ; Un H and so on as in Art. 447. 2. Put -+ * ^ then (a, + j/^) (a^j + x)= 0,0^1 ; whence «, = ^]^__ . and BO on as in Ex. 1, Art. 447. 3. Let —-3 = — — ; then Xi= — y; replacing r by »•+ 1, we have r r+1 r+1 5 = ; , where x„ = . r-1 r+l-Xa " r Similarly = — — , where x, = ; and so on. r r + 2-x^ ^ r + 1' 4. We have r = = , where x, — -^ — ; 71 + 1 1-Xi ' 2n n-1 1 , 2 (re- 2) -j5 — = ;; , where yi = — — =— ; 2n 4-2/, "^ n-1 ' replacing re by re - 2, we have 2(re-2) 1 , n-3 1 ■ , and re-1 re-3 ' 2(n-2) 2(re-4) ' 2(re-2) * re-3 Moreover since the numerators in the fractions re-1 2 (re - 2) re-3 2(re-4) 2re ' m-1 ' 2(re-2)' re-3 ' *" diminish by unity, there will be re components on the right. 5. We know that l- + l + i+ +1 1 M," «,2 uj On putting 1*1 = 1, «2=2, ^3=3, ...we obtain the result. H. A. K. 13 194 CONTINUED FRACTIONS. [CHAP. 6. In the equation of Ex. 5, on putting we obtain the result. 7. We have «''=1 + I + p" + ^ + ^ + - In Example 2 on putting a„ = l, iii = l, a2=2, 03=8, ..., the result follows at once. 8. Let r= ~; thus(6-l){a + /3) = a6; and B = r^ ; "a aU a+ b-1' „■ -, , 11 1 1 1/1 IN 1 1 Similarly - + -— = r-r-= r, ^ . suppose, a ab abc a a\b be) a 0(6 + 7) la lab , = — r — :; = z — = ^ ; and so on. a+ 6-1 + 7 a+ 6-1+ c-1 9. From Art. 447, we have 1111 1 V tt 2 M 2 — + — + — + — + . ^ ^- 5 «1 U., % 1(4 «!— tti + Mj— Kj + ^a" "s + ^J" 1 1 1 1 1 T^ r8 r'8 J. + ^1 "*" ^ ''' ,.1B ,. _ J. + )J _ )-4 + j-9 _ ^9 ^_ ^6 . on reducing as explained in Art. 448 we have the result required. 10. This is an easy consequence from Art. 448. Thus a^ a.^ «3 1 dt^ '^I'^s 1 1 ^^3 1 1 ^1 «!+ 05+ aj 1+ a^a^+ a^ 1+ a^+ a^Ug 1+ ^1+ a^' 11. We have P=^ * ^^... ^'- a+ b+ c+ d + _ 1 b ac d ~1+ ab+ c+ d+ "' _ 1 1 ac bd ~T+ oT bc+ 'd+ '" = ■ -— — ... [Compare Art. 448.1 1+ a+ 6+ c+ L -f ■> 1+ a+Q _ a + Q a+l+Q' XXXIl] PROBABILITY. 195 12. From Ex. 2, Art. 447, we have 1 X t' x^ 1 a,^x a^x a."-x + +...= - Oj + Oj — a-^x + Oj - a^ + 04 - a.^x + Hence 1 1-... 1\ 2l?2 9233 tzii 1 q-^x 9i+ 3i32-9i^+ 9273-3i92^+ M4 -9293^ + _ 1 X X a "91^92^^ 93-91^ I 94-9yC , ■" 9i 92 93 \ X X X «!+ «2+ a3+ 14 + since 9i=%; 92="i<^2 + ^; 93 = "392 + ^9i ! 94= that is, the sum of the odd coefficients in (1 + 1)" divided by 2". 2n-i X .'. the required chance = -^ = -= , 6. Two coins can be drawn in 10 ways, and two sovereigns in 1 way ; .-. ^'s expectation on this ground=Y^ x 40=4 shillings. One sovereign and one shilling can be drawn in 6 ways, and A'& expecta- 3 tion on this ground=,= x 21 = 12f shillings. Two shillings can be drawn in 3 3 3 ways, and A's expectation = == x 2=- shillings. .". on the whole A'& expectation = 17^ shillings. Or more simply thus: 2 The probable value of a draw is ^ of the amount in the bag. 2 /. A's expectation = J of 43 shillings = 17 J shillings. XXXII.] PROBABILITY. 203 7. In his first throw the fourth man's chance is r^ , in his next throw '* ^^ 2" ' °'°'^ ^° °^ ' *^^''^f°''^ Ws chance is the sum of the infinite series 24 + 2" "'■^iS'*'"" Thus the chance ^^.^fl-K) = ^^ 2* \ 2«J 63 8. The required chance is ohtained by dividing the coeESeient of x^ in Now (a; + a;2 + x3)3=x3(l + x + a;2)3=33 Qz^Y ; we have therefore to find the coefSoient of x' in (1 - x^f (1 - x)-''>, that is, in (l-3x3 + 3a;6-a:S) (l + 3x + 6x^ + l{)x^ + ...). This coefficient =-3 + 10 = 7. .". the required chance =^ . 9. If the sum of the numbers is less than 15, the numbers must be 3, 3, 3, 3 or 3, 3, 3, 5. And in this last combination of numbers the 5 may occur in any one of the four throws ; thus there are 5 cases favourable, and 16 cases in all. 5 .•. the chance required = -^ , 10. The three dice can be thrown in 216 ways. The number of ways in which the dice can be thrown so as to have a total of 10 is the coefficient /I — a'N' of x'o in (x + x2 + a;'+... + x^)'; that is, in x' ( ^j ) ; now this is the same as the coefficient of x' in (1 - x^)^ (1 - x)-^, and is found to be 27; 27 1 .•. the required chance = rrr; = = . ^ 216 8 11. In order to win the set, B must win 2 games before A wins 3. Therefore by Art. 466, B's chance = (l)' il + 2 . ^ + ^ (-Y\ , or — . \2y ( 2 1 . 2 \2J J 16 .•. B'a share=£ll, and A's share=£5. 12. The number of ways in which the 3 dice may fall is 6', or 216. In order to lose, B may throw anything from 3 to 8 inclusive; the number of ways in which this may be done is the sum of the coefficients of the powers of x from 3 to 8 inclusive in the expansion of (x + xHa^ + .-. + xS)*. 204 PEOBABILITT. [CHAP. ——1 =a;3(l-a;6)'(l-a;)-' =a;5(l-3a;8 + 3xi2-a;i8)(l + 3x + 6a:» + 10x= + 15a;* + 21a;= + ...). Thus the number of way3 = l + 3 + 6 + 10 + 15 + 21 = 5G; 56 7 hence the chance that B loses is ^rrr. = ;^ • 216 27 4 2 13. The chance of drawing a sovereign in the two coins is tt^t , or = . 10 6 12 21 In this case G'b expectation = - x 5 of 21 Bhillings = — shillings. The chance that both coins drawn are shillings=-, and in this case C'a 6 13 3 expectation = 5 x = of 2 shillings =j of a shilling. Thus the whole expectation =— = 44 shillings. o Or more simply thus: 2 The probable value of any two coins =5 of 24 shillings; and C's expecta- u tion is half of this sum. 1 «j 14. With the notation of Art. 462, we have p=%, g=si n—5; hence 6 6 250 ""^TTTe- the chance of throwing exactly three aces is 'Cj •(a) • ( S ) ' The chance of throwing three aces at least is 276 Thus the chance is ===;; . 7776 15. The chance of throwing 7 with two dice is - , and the chance of 6 throwing i ia^r^. Thus ^'s chance in each trial is double of B's. Now we require B's expectation in the long run, the throwing being con- tinued until one or other of them wins. Let x = B'b chance on this supposition, then clearly 2x=A'b chance, and therefore x + 2x = l. 1 12 Therefore x=-^, and B's expectation = 5 of 5s. - 5 of 2s. = id. o ■ ■ o o XXXII.] PROBABILITY. 205 16, The two dice may be thrown in 4 x 6 or 24 ways. The numbers of ways in which 2, 3, 4,... 10 may be thrown are given by the coefficients of those powers of x in the expansion of {x+x' + afl+...+x^) {x + x^ + x' + x'^). In the question before us, the required event will happen if any of the numbers from 5 to 10 inclusive be thrown. Thus the required chance -A A i. i. A JL-1?-? ~ 24 "'' 2i ■•" 24 "*" 24 "^ 24 "^ 24 ~ 24 ~ 4 ■ 17. Let the purse contain n coins in all. Then the expectation from the first draw is - (M+m). n-1 Now the chance of a second draw is ■ , and here it is certain that M n remains, also n - 2 other coins, each of which has an average value =^ ; M-2 their total value is therefore z- . m ; n-1 .•. the expectation from 2'"' draw= . =- \2I-\ mi- re n-1 ( ra-1 J = - \M+ =m\ . n [ ra-1 ) Similarly the chance of a 3'* draw is . = , in which it is certain n n— 1 that M remains and n-S other coins of average value — — j . n~l ra-2 1 (,^ re-3 ) .•. the expectation from 3"^ draw= . =- . s i^+ r • "^r '' n n-1 n-2 [ n-1 j n [ n-1 i and so on; the expectation from the last draw being - {M+0}. .: the whole expectation ■1 /I 1 . \ m fi "-2 re-3, . — ^, 1 = jl/(± + _ + ...ton terms I + - {1+ — = + =■ + ... to ra-1 termsV \n n / re ( n-1 re-1 j ,, (ra-l)re m 1 ,, , 1 2 re re-1 2 [This problem and solution are due to the Bev. T. C. Simmons, M.A.] 206 PROBABILITY. [CHAP. 18. Tlie total number of ways in which three tickets may be drawn is 6n(6n-l)(6n-2) ,„ ,,,„ „. X.2 T3 ' ~ ' ^ " ^' To find the number of ways in which the sum of the numbers drawn is 6n we may proceed as follows : First suppose is drawn, then we have to make up Gn in all possible ways from two of the numbers 1, 2, 3, ... 6re-l ; this can be done in 3n-l ways. Then suppose 1 is drawn ; we have to make up 671 - 1 from two of the numbers 2, 3, ... 6m-l; this can be done in 3m-2 ways. If 2 is drawn, we have to make up 6it — 2 from two of the numbers 3, 4, ... 6m- 1; this can be done in 3;i-4 ways; if 3 is drawn, there are 3m - 5 ways of making up the number. Finally, if 271-2 is drawn, there are only two ways of making up the numbers, viz. 2n-2, 2m-l, 2n + 3, and 2«- 2, 2n, 2re + 2; while if 2k-1 is drawn, there is only one way, viz. 2» - 1, 2n, 2re + 1. Hence the number of ways of making up 6n is the sum of 2n terms, which may be arranged in n pairs as follows : {(3n-l) + (3n-2)} + {(3n-4) + (3n-5)} + ... + (5 + 4) + (2 + l) = (6re - 3) + (6n - 9) + (6n - 12) + ... = 3n\ Thus the required chance =.3ji'^-j-n (6n - 1) (6n - 2). EXAMPLES. XXXII. d. Pages 399, 400. [Where the wording of a question admits of two interpretations, as in the Example on page 305, we have here adopted the first method of solution there explained.] 1, There are four equally likely hypotheses, namely, the bag may have contained 4 white balls, or 3, or 2, or 1. 3 2 1 And i>i=l. Pa=-g, 1'3=J. Pi=l- Thus the required chance ==^l-r = —- = -. S [pj 10 o 2. The four hypotheses here are 6 black balls, or 5, or 4, or 3, and these are all equally likely. 543 432 321 And ft=l, y2=6-5-i. ^3=6-5-4' ^'^Q-l-p . Pi_P2_P3_n_'^p) •• 20 10 4 1 35 ■ .•. the required chance = ^f-/^ = Tr= . Z [PI 00 XXXII.] PROBABILITY. 207 3. If the letter came from Clifton, there are 6 pairs of oonseontive letters of which ON is one. ■ Therefore the chance that this Was the legible couple on the Clifton hypothesis is ^ . If the letter came from London, out of 5 pairs of consecutive letters 2 are ON. Therefore the chance that this was the legible couple on the 2 London hypothesis is ^ . Therefore the a posteriori chances that the letter Was from Clifton or London are 1 2 — — , and respectively. e'^S 6''"5 Thus the required chance = z-=. 4. A could lose in two ways ; either by B winning or by winning. 3 2 The probabilities of these two events are ^, ^ respectively. Therefore 5 1 A'b a priori chance of losing was :^a, or -, But after the accident his 2 chance of losing becomes ^ ; that is, his chance of losing is increased in the ratio of 4 to 3. Therefore, also, B'a and C'a chances of winning are 3 4 2 increased in the same ratio. Thus B's chance of winning = ^j-^r x 5 = r 5 ^^^ 2 4 4 C'b chance of winning = ^j; x ^ = y^ . 5. There are n equally likely hypotheses, for the purse may have con- tained any number of sovereigns from 1 to n. 12 3 re Thus pi = -, p,=-, Ps = ~ Pn = -. 2 .■. the required chance = .^y— = X(p) n(n + l) 6. There are two cases: either the coin had two heads, or it had a head and a tail. Thus Px=^, A=^. Also i'i=l. ^2= (2) ' 208 PROBABILITY. [CHAP. Therefore | = T = n- 32 Thus Qj=— =the required chance. 7. We have five oases to consider, for the bag may contain 1, 2, 3, 4, or 5 red balls, and we suppose these to be all equally likely. Hence i'i=Qy. P2=(ff, Ps={i)\ i'4=(ty. i'a=(5)'- • • 12 ~ 22 32 42 52 55 ■ The chance of now drawing two red balls -55 15 + 10+ 5 +^7 "550- 8. See Case U. in the Example to Art. 473, whence it appears that the chance of 5 shillings is ( 5 ) , of 4 shillings ^ , of 3 shillings — , of 2 shil- lings 25- 5 thus Pi 1 ~32' , F2 = 5 32' -P3= 10 '32' p_10 Also i'i=l. ^"2 = 4 3 =5^4^ . Ps 3 2 -s'^i' 2 P4=gX 1 4" •■ • PiPi= 1 '32' J'A = 3 ^82' ^3? , 3 '~32 . PiPi = 1 "32 . Qi_Q3_Q3_Qi_^ ■"1 3 ~ 3 18" Hence the probable value in shillings of the remaining coins 1„353„139-, , .„. = 8^^ + 8 " 2 + 8^2 + 8 '^2 = 4=^i'^^'^"§'- 9. Beckon the result of the last two throws in one total. Then the whole throw of 15 can be made up as follows: 3 + 12, 4 + 11, 5 + 10, 6 + 9; and these four cases can occur in 1, 2, 3, 4 ways respectively, all of which are equally likely ; • ^=^2=^3 = ?!. "1 2 3 4 ' _^2__A-i XXXII.] PROBABILITY. 209 10. Denote A's and B's veracities by p and p', then the required probability is p(i--p')+P' 0-~ P) i ,, , . 3 15 18 1 that 18, -. — I — . — = — = -■ 4 6 6 4 24 3 11. There are two hypotheses; (i) their coincident testimony is true, (ii) it is false. With the notation of Art. 478, we have p_l p_5. ■^1-6' A-6' _2 4 _1 1 1 for in estimating p^ we must take into account the chance that A and B will both select the red ball when it has not been drawn. Thus PiPi •■ -P22'2=8 . ^=40 : 1 ; . 40 hence the probability that the statement is true is jy . 12. The antecedent chance that the lost card is a spade is - , because 3 there are 4 suits ; and the chance that it is not a spade is j , Thus -.4. ..=l. also 12.11 ^1-51.50' 13.12 ^2- 51 . 50 . Ci_ «2 _ 1 ■"11 3x13 50" .•. Qj = — = the chance that the missing card was a spade. 13, There are three hypotheses ; A may have won £5, £1, or nothing, for B and G may both have been mistaken. Thus P, = l, P,=^, P3 = ^. 2 3 Since B'b veracity is represented by ^ , and C's by j , we have 2 1 13 11 •■ 2 ~ 3 8 13* 2 3 Thus A's expectation =Yo of £5 + 73 of £1 = £1, H. A. K. 14 210 PROBABILITY. [CHAP. 14. There are three equally likely hypotheses ; for the purse may cou- tain 2, or 3, or 4 sovereigns. Now ^=6' -Pa^^i' ^3^^' p 3 .'. the chance that all are sovereigns = ;^ — ^ = - Arrain Pi _P2 _P3 _'^jP) Again ^ _ g _ g _ ^^ . 3 -X{p) 5- •■• ^i-Io- ^2-io' ^3-io- .". the chance that another dra-wing will give a sovereign 15. At first, B'b chance of winning his race is - ; similarly C's chance is - , and D's chance is - . o o Therefore after the 2""' race is known to have been won by B or D, B'a chance : certainty : : ^ : u ; 3 that is, B'a chance=7- . o 3 11 Therefore, the chance of P winning his bet = 1x5X5 = -; and the chance 8 3 8 7 of P losing it is - . o 1 7 Thus P's expectation =3of£120-3of£8 = £8. 8 16. We have n cases to consider, for there may be 1, 2, 3, ...m white balls; and all these cases are equally likely, so that Pi=P2=Pi=...=P„. If there were r white balls, the chance of drawing two white balls in this case would be ( - | . ._2l=-22_= =_2£.- 1 erer © ■■°-(5)' 6re 6r^ (n + 1) (2,1 + 1)' ^°'''^'—«(,H.l)(2re + l)- Thus -ii? = — — — , and Q^= ©" XXXII.] PROBABILITY. 21 1 And the chance of another drawing giving a black ball "n-r 6r2 BSr* 62r' = S ,-1 n ■n(?i+l)(2n + l) n (n + 1) (2/1 + 1) n^ (n + 1) (2n + 1) , fa" (" + !)" 1 3(n + l) 1, ,,,„ ^^, , = ^- 4n''(«+l)(2«+l) = ^-2(2;r4 = 2'"-^»^" + ^' " 17. Eepresent the two coins by A and B. Then the o priori chance n-1 that B is with 4 is ^ , for wherever 4 is placed, there remain mn - 1 possible positions for B, « - 1 of which are favourable. Hence the a priori chance that B and A are not together is — — ^ . mn-1 Now consider the m — r purses which have not been examined. If A and B are together, the chance that they occur in these purses is . If A m and B are apart the chance that they both occur in these purses is (m-r)(m-r-l) , , ,> . ., . . , , „ . , . , , ^ -. =-r ; for m (m - 1) IS the total number of ways in which they m(m-l) ^ ' ^ ■' can occur separately in any two purses whatever, and (m - r) (m - r - 1) is the number of ways in which they can occur separately in any two of the purses we are considering. Hence the required chance _ n-1 m-r _ \n-l m-r n{m-l) (m-r)(m-r-l)) mn —1' m ' (mn — 1 ' m mn — 1 ' m{m — l) j mn -nr — 1' 18. The chance that A and B both get the correct result is 5.7;;; the o 12 7 11 chance that they both get an incorrect result is 5 . 77; ; and therefore the o 12 chance that they get the same incorrect result is :j-t-- • 5 • ttt = To — q - lUUl o 12 lo . o . 12 Thus the chance that their solution is correct is to the chance that it is incorrect as 1 to — , or as 13 to 1. 19. Let p be the a priori probability of the event; then the probability that their statement is true is to the probability that it is false as (|)"'^isto(l-:p)(iy Therefore - — — represents the odds in favour of the event. Now in order 1-p that the odds in favour of the event may be at least five to one, we must have = — — not less than 5 ; that is, 5^p must he not less than 1 -p, or (5'+l)^ must be not less than 1. Hence p must be not less than ^ — -. 5^+1 14—2 212 PROBABILITY. [CHAP. EXAMPLES. XXXII. e. Page 405. 1. By writing down the different combinations it is easy to see that 12 can be thrown in 1 way, 11 in 2 ways, 10 in 3 ways, 9 in 4 ways, 8 in 5 ways, 7 in 6 ways. Therefore out of the 36 possible ways of throwing the dice there are 1 + 2+3 + 4 + 6 + 6, or 21 ways favourable to throwing 7 7 or more. Thus the chance of throwing at least 7 is ^5 • 2. The nine coins can be arranged in [9 ways; but the five sovereigns can be arranged in the odd places and the four shillings in the even places in 1 5 X 14 ways. Hence the chance that they wiU be drawn alternately |5x|4 1 beginning with a sovereign is '— „ = r^r^ . Or thus: The number of ways in which nine things can be arranged, when five are alike of one sort, and four are alike of another sort, is ■>'—.. , or 126, and all these ways are equally likely. .•. the required chance = :r-r;; . 3. See XXXn. b. Example 20. 4. The first person's chance is - ; if he fails, since there are n - 1 tickets left, the second person's chance is - = - . If the first two n n— 1 n fail, the third person draws from m - 2 tickets, and his chance is m-1 w-2 1 _1_ n ' n—1' n-2~ n' and so on. Thus each person's chance is -. n 5. The chance that the first bag is chosen is - ; and the chance of choosing one white and one red is 5 x 3-^^02. Again the chance that the second bag is chosen is - , and the chance of choosing one of each colour is now 4x5-i-'(7j. ,, ■ , ■. 1 15 1 20 275 .•. the required chance =-.T:;r + -.-— = . 2 28 2 36 504 6. ^'s chance = 6 T"*" (g) "^ (?) +—| = S suppose; — =|.iji+gy+(|yv..,}=|.. XXXII.] PROBABILITY. 213 Similarly C's chance is ( x J S, while D's chance is ( - j S, and E'b chance is ( p J S. Thus their respective chanees are as 7. Three squares may be chosen in ^C^ ways ; two white and one black, or two black and one white may be chosen m '^Cj x 32 ways. ™ ,, • J V 2x32C„x32 16 Thus the required chance = ^ = — - . i^Cj 21 8. The two dice may be thrown in 24 ways. The number of ways in which 2, 3, 4, ...10 may be thrown respectively are given by the coefficients of those powers of x in the expansion of {x + x^ + x^ + ...+x<')(x-i-x' + x^ + x*). Multiplying out, we get x^ + 2x3 + 3x* + ix^ + 4x< o = sr ■ Similarly the chance that A and B are adjacent, 2 A being at the end of the carriage is -^ . These events are mutually ex- elusive ; hence the whole chance of A and B being adjacent = ^ . (2) may also be solved as follows : The total number of pairs of positions in which A and B can be adjacent = 4. The total number of pairs of positions they can occupy without restric- tion, inside or outside, is '(72 = 21. 4 .•. the required chance = ^ . 15. In order that a number may be divisible by 11, the difference of the sum of the digits in the odd and even places must be either zero, or a multiple of 11. [See Art. 84.] Here the difference cannot be zero, so that we have to divide 59 into two parts whose difference is 11 ; these parts are 35 and 24. But the suin of three digits cannot be equal to 35; hence the seven digits must be such that the sum of the four odd ones is 35, and the sum of the three even ones is 24. Now the number of ways in which 7 digits may be arranged so as to make 59 is equal to the coefficient of a;^' in the expansion of (a;'' + a;' + j;^+ ... -vx^Y ; and since the coefficients of terms equidistant from the beginning and end are equal, this is equal to the coefficient of ic* in the above expansion ; that is, is equal to the coefficient of a;* in the expansion of (1 - x")' (1 - xy. This coefficient is 210. Again, the number of ways in which 4 digits can be arranged so as to make 35 is equal to the coefficient of x^^ in the expansion of (a;i>-|-a;i+x'i+...-l-a:Y, and is therefore equal to 4. Similarly, the number of ways in which 3 digits can be arranged to make 24 is equal to the coefficient of x^ in the expansion oi (x^ + x^ \ x^ ^^ ... +a;')', and is therefore equal to 10. Each way of arranging the odd digits may be associated with each way of arranging the even digits; 4 X 10 4 hence the required chance = -^Y^r- = nj . 216 PROBABILITY. [CHAP. 16. The number of favourable oases is the coefficient of x^^ in the expansion of x^ (1 - x^)^ (1 - x)"^. [See Ex. 2, Art. 466.] Putting this in the form x3(l-3a;« + ...)(l + 3x + 6a;2 + 10a;3 + ... + 55a:9+...), we easily see that the number of favourable oases is 55 - 30, or 25. 25 25 Thus the required chance = -^ = ^r^ . 17. The total number of drawings is 1*. The number of ways in which the sum of the drawings will amount to 8 is the coefficient of x' in the expansion of (x° + x^ + x^ + . . . x^)*. This expression = (l-x'')*(l-x)-^=(l-4x'+...)(l + 4x+... + 165x8+...). Thus the coefficient of x^ is 165 - 16, or 149, and the required chance is 149 2401' 18. (1) We must find the coefficient of x^" in the expansion of (x" + x" + x" + x" + x" + x^ + x'-^ + x' + x^ + x')3 and divide it by 10'. Put y for x + x^ + x' + x^ + x'; then (5 + yY = 5^ + S . 5Hj + 3 . 5y^+y^. The coefficient of x^" comes from the last two terms only, and is equal to 33 15 + 18. Thus the required chance = tttxt; . (2) There are now two favourable cases, namely those in which the tickets 1, 4, 5, or the tickets 2, 3, 5 are drawn. And the whole number of cases is ^"Cj, since the chance is just the same as if the three tickets were 2 1 drawn sunultaneously. Thus the required chance = .-=j; = ^ . 19. (1) If the last digit be 1, 3, 7, or 9, none of the numbers can be even or end in or 5 ; that is, we have a choice of 4 digits with which to 4" /aN" end each of our re numbers. Thus the required chance = -^ =(•= I . (2) If the last digit be 2, 4, 6, or 8, none of the numbers can end in or 5 and one of the last digits must be even. Now 8" is the number of ways in which we can exclude and 5 ; and of these we have further to exclude the 4" cases in which the last digit can be selected solely from 1, B, gn _ 471 ^n _ 2» 7, or 9. Thus the required chance = = — ^ — . (3) If the last digit is 5, one of the numbers must end in 5 and all the rest must be odd. Now 5" is the number of ways in which an odd digit can be chosen to end the number, but to ensure 5 being one of them we must exclude the 4" ways in which an odd digit can be chosen solely from 1, 3, 7 or 9. 5n_4n .•. the required chance = — -— — . ^ 10" (4) We have now to subtract the sum of the previous chances from unity. XXXII.J PROBABILITY. 217 20. This is a particular case of Ex. 17, XXXII. o. and may be solved in the same way. Or we may proceed as follows. If the dummy is drawn first the value of the draw is nothing. If it is drawn second, the value of the two draws =i(£l. + £l. + ls. + ls.) = 10s. 6d. If it is drawn third, the value of the three draws =g(£2. + £l. ls. + £l. ls. + £l. ls. + £l. ls. + 2s.) = £l. Is. If it is drawn fourth, the value of the four draws = j(£2. Is. + £2. ls.+£l. 2s. + £1. 2s.) = £1. lis. ed. If it is drawn fifth, the proceeds of the five draws =£2. 2s. All these cases are equally likely ; hence the whole expectation =i(0 + 10s. 6d. + £l. ls. + £l. lis. 6(j'. + £2. 2s.) = £l. Is. o 21. The chance of throwing 10 with 3 dice is g. [See XXXII. u. Example 10.] A throws first, and the chance that JS has a throw is - . So that if x be o A's chance of winning, B'a chance is jx, and C's chance is ( g j x. And the sum of these three chances is 1, since they continue throwing imtU the event happens. 22. The solution of this Example is exactly similar to that of XXXII. d. Example 11. 23. The chance of drawing the single counter marked 1 is ^, ^j -r ; the 4 chance of drawing one of the two counters marked 4 is ^, _^^, ; the chance 6 , of drawing one of the three counters marked 9 is — r- ; and so on. .-. the required expectation in shillings n(n + l)' -i 218 PROBABILITY. [CHAP. 24. The number of ways in whicli a man may have all 10 things is 1 ; the number of ways in which he may have 9 things is 10 x 2, for '"0^ = 10, and in each case the remaining thing may be given in 2 ways. Similarly 10 . 9 he may have 8 things in " . 2^ ways, for aiter taking away a combiQation of 8 things the remaining 2 may be given in 2^ ways. Similarly a man may have 7 things in " 2' ways, and he may 10 9 8 7 have 6 things in ' . 2* ways. And the total number of ways, in which 10 things can be given among 3 persons is 3^°. .". the chance of a man having more than 5 things _ 1 + 20 + 180 + 960 + 3360 _ 4521 _ 1507 ~ 3" ~ 3" "19683" 25. Let the rod be divided into n equal divisions A^A^, A^Ag, A^A^ and let the random points of division be denoted by Pj, P^, F^, .... Then first it is necessary that one of the random points falls in each division ; the Ire chance of this is ■!= , for the total number of cases is the number of ways in re" ■' which II. places can be occupied by re things when repetitions are allowed, and the number of favourable cases is the number of ways in which n places can be occupied by re things when repetitions are not allowed. Again AiP^ must be greater than A^P^, or PiP^ would exceed - of the rod; therefore A^P-^, A^P^,... are in descending order of magnitude. The chance that this particular order will occur is t- , for the number of orders '- . in which they can occur is Ire, and all are equally Ukely. Thus the required chance = — - . re" 26. Denote the two purses by B^ and JJj. Four cases are a priori, namely, (1) a sovereign may be transferred from JSj to B^, (2) a shilling ^i to 732, (S) a sovereign B^to Bi, (4) a shilling B^toB^. Then since the chance of drawing from either purse is 5 , we have p _1 3 p _1 1 p _1 1 D _1 3 J^i-2^i'' ^~2^i' 3-2^4' ^~2^i" XXXII.] PROBABILITY. 219 In (1), Bj has 2 sovereigns, 1 sliilling, B^ has 2 sovereigns, 3 shillings; tT, + 13 1 BO that2)i=-x- = -. In (2), J3j has 3 sovereigns, B^ has 1 sovereign, 4 shillings; so thatp2=0. In (3), B\ has 4 sovereigns, 1 shilling, B^ has 3 shillings; so that ;,3 = -xl = -. In (4), Bj has 3 sovereigns, 2 shillings, B^ has 1 sovereign, 2 shillings; .. ii i 2 2 4 SO that ^4=5X3=15. •■• ■Pii'i=4o; -P2P2=o; -^3^=40; -^4^4=4^- ...| = ^4^.henoe,,4. Now for the second trial we have only to consider the case which corre- ■ sponds to Q^, for in none of the other cases could a shilling be drawn from each purse. .•. the required chance = QiX-rX= = r-^. 4 2 lb 27. Draw tangents to the circle at the three random points, thus forming a second triangle. Then if the first triangle is acute angled, the circle is inscribed iu the second ; and if the first triangle is obtuse angled, the circle is escribed to the second. Hence the required result follows as in Ex. 3 of Art. 481. [This problem and solution are due to the Eev. T. C. Simmons.] 28. Let A, B, C he the three points; then in favourable eases the sum of any two of the angles of the triangle ABC must be greater than the third. That is, the triangle must be acute angled, and by Ex. 27 the chance of this is J. [Eev. T. C. Simmons.] 29. Let AB be the straight line divided at P and Q; let AB=a, AP=x, BQ=y. ^ p g g Then the favourable cases require .-. a-{x + y)<^, or x+y>^. And the possible cases require x + yca. 220 PROBABILITY. [chap. a 2' ^ = 2- Tate a pair of rectangular axes 00, OD; let OG, OD be each equal to a, so that CD is the line x + y = a. Bisect OD, 00, OD in E, F, G respectively. Then GF is the line x+y=~; and GE, EF are the lines y = Now the favourable cases are restricted to points in the triangle EGF, and the possible oases include all points in the triangle OGD. Thus the required chance= j. Or thus: If the 3 parts of the line are x, y, z we must have x + y+z = a, while x + y>z, y + z>x, z + x>y. Therefore x, y, z must each be <-. Therefore if we take three rectangular axes OA, OB, 00, and make 01, OB, OG each equal to a, the plane x + y + z = a includes the points which give the possible oases, while the favourable cases are restricted to the triangle DEF, where D, E, F are the middle points of BO, CA, AB respectively. Thus the required chance=j. 30. Let Pi, P2 be the a priori probabilities of drawing i sovereigns from the 1" and 2'"' purses respectively. Then p^ = l, and p,=ioC4-=-2=C,=j|ig. •"1265 21 1286" •• ^'~1286' ^''^i286* Again the probable value of the next draw in pounds is 31. Let AB be the straight Une of length a, and let the random points P, Q be at distances x, y from one end of the line. Now in favourable cases we must have x>b + y, or y>b + x^ A Q B XXXII.] PROBABILITY, 221 Again in possible oases we must have x>0 and 0 and . Now the favourable cases are restricted to points within the triangles CEF, GDH, while for possible cases we may have all points in the figure CD. Thus the req^uired chance : ■C-^T 32. In the line AB let points P, Q be taken in the order APQB so that AP=x, BQ = y, PQ = a-x-y. Then in favourable cases we must have x<:6, j/<6, a-x-yg, the favourable cases will be restricted to the shaded area in Fig. 1, and fa-b\' the required chance = 1 - 3 I I . (2) When 6<^. tlie favourable cases will be restricted to the shaded area in Fig. 2. This consists of a right-angled isosceles triangle each H of whose siie3 = 0F - EG=b -(a-2b) = 3b - a. F .•. the required chance = I — — I . 222 PROBABILITY. [chap. 33. Let AB be the line of length a + b, and on it measure AP=x, PQ = a; also let 4R=y, ES=b. Then for possible cases we must have x>0, and <6; y>0, and and < 6 + c, and y>0 and <.a + c. Also in favourable cases we must have (1) AQ-AP'-^d'i or (2) AQ'-AP<:d\ x + a-y'idK b + y-xa—d) x—y>b~d] Talie a pair of rectangular axes 0(7, OB. Let 0C = 6-l-c, OB—a + c, OE=b, EF=d, GE = c. Also let OE =a, E'F' = d, E'B=c. XXXII.J PROBABILITY. Then OF'=a-d, OF=b-d. 223 A P Q B P' Q' B Draw the line y-x = a-d represented by LF'T in the figure, and the line x-y = b-d, represented by MF. Then DF'=DL = CM=CF=c + d. And the favourable eases are re- stricted to points in the triangles GFM, DF'L, whUe the possible cases include all points in the rectangle 00, OD. Thus the required chance = , r-, rr • {c + a)(c + b) 35. The chance that G travels first class =■ I the chance that A ■ ; therefore the l + m + n' travels in any particular first class compartment = y . chance that C and A travel together, both in the same first class compart- ment = ,5 rr. ^ . {l+m + n){\ + ix. + i>) The chance that A and C travel together both in the same second class u, compartment = -; f-r ^. . Similarly for companionship in any the same third class compartment. Thus the chance of A and C being companions iu some compartment _ 'K + fi + v 1 " (l + m + n) {\ + /J. + v)~ l + m + n' 1 Hence the chance that A is with C, and B with D = chance that A is with one lady and B with the other is (l + vi + n)^ 2 ; and the (l+m + n)^ Again, the chance that A and B both travel first class = t X" , and (X-H/i-t-^-P the chance that they both travel first class in the same compartment = ~ ^' ~r (x+f^+i^y Thus the whole chance of their travelling together in some compartment 1 \''mn + fi-nl + ;'^te \ I m, n J (X-h/i-l-cJ^ Imnl^ + ii + vY 224 PROBABILITY. [chap. Now C's ehanee ie the same for every compartment ; therefore the chance that A and B are together and C also in their company _ 1 \^mn + fi?nl + vHn _ l + m + n' Imn (X + /* + vj^ and the chance that A and B are together and one or other of the ladies with them is the double of this, or 2 "Khnn + fi'nl + vHm l + m + n' lmn{\ + fi + v)^ o We have to prove that this is greater than ■ (l + m + n)''' This will be the case if (X^mre + ij,^nl + vHm) (l + m + n)> Imn (X + /i + v)^, that is, if I {/j.^n^ + v V) + m {vH^ + \^n^ + n {X^m" + jxH^) > 2lmn {/j.v + v\ + X/i), an inequality which always holds except when I : m : n = \ : /j. : !>. [This problem and solution are due to the Eev. T. C. Simmons.] EXAMPLES. XXXIII. a. Pages 419 to 421. 1. Subtracting the first column from the second, and also from the third, we have 1 1 1 = 35 37 34 23 26 25 10 35 2 -1 23 3 2 = 12 -1 1=7. Is 2 2. Adding together the first and last columns, we obtain a column in which each of the constituents is double of the corresponding constituent iu the middle column ; hence the result is zero. 3. Keeping the second column unaltered, first multiply it by 4 and sub- tract from the first column; then multiply the second column by 7 and subtract from the third ; thus 13 3 23 30 7 53 39 9 70 13 2 = 2 7 4 3 9 7 13 2 7 3 9 1 = 11 3 1 = 1. I 2 7 I 4. Here 5. Here 6. Here a h g h b f g f e 1 z -y — Z 1 X y -X 1 = a{hc-f)--h{ch-fg)+g(fh--bg). 1 (l + x^)~z(-z~xy)-y[xz-y). 111 1 1 + x 1 1 1 1 + 2/ 10 1 ic 1 y = xy. XXXIII.] DETERMINANTS. 225 7. Adding together all the columns we obtain a new determinant in which all the constituents of one column are zero ; hence the value of the determinant is zero. 8, Add together the second and third row and subtract the sum from the first row ; thus b + c a a b c + a b c c a + b -2c -26 6 c + a b c c a + b = 2cb{a + b-c)- 26c (6 - c - a) = 4o6c. 9. Adding together all the columns we obtain a new determinant in which all the constituents of one column are 1 + w + w^, that is, equal to zero ; hence the value of the determinant is zero. 10. Since u^ is equal to 1, we have 1 W3 a' = 1 1 0,2 = 1 0,3 1 03 1 1 0, 1 w2 a 1 u" Ct, 1 0,2 0,-0,2 = _(u_u2)(o,-0,2)__„2 + 2o,3-U*=2-(o,2+0,) = 3. 11. The result of the elimination is a c 6 c b a b a e = 0; that is, a (6c-a2)-c(c2-a6) + 6(ac-6') = 0. am, e have a I c = - X y z = + X y z . X y z a b c p q r p q r p q r u b c a 6 c u X p = - b y 9 = + y b q X y z b y q a X ■P X a p P 1 r c z r c z r z c r 1 13. (1) The given equation is a quadratic, and clearly vanishes when a;=a, or when x=b; hence the solution is x = a or 6. (2) Add together the first and second rows, and subtract twice the third row from the sum; thus 15 -2x 11 10 11 -Sx 17 16 12 - 3a; =0; hence (12 -3x) l 11 10 i =0; therefore 12-3x = 0, and a: = 4. I 17 16 I H. A. K. 15 226 DETERMINANTS. [chap. 14. Erom Art. 495, this determinant can be expressed as the sum of eight determinants, all of which vanish with the exception of h c b c a and c a b 1 r p r ■P 1 y r. a; z X y • each of which is equal to p q r X y z 15. This determinant vanishes if a=b, and therefore must contain a-b as a factor ; similarly it contains b-c and c - a as factors ; and there- fore being of the third degree must be equal to k{b-c) (c-a) {a-b}. By comparing the coefficients of bc^, we see that fc = l. 16. As in Ex. 15, the determinant is divisible by (b -c){c- a) (a - 6), and being of the fourth degree the remaining factor must be k [a + b + c). By comparing the coefficients of 6c', we see that Je=l. [See Ex. 2, Art. 522.] 17. As in Ex. 15, the determinant is divisible by {y -z) (z- x) (x-y); the remaining factor must be of the form A (x^ + y^ + z^)+B (yz + zx + xy). Since the highest power of x in the determinant is n?, a comparison of the coefficients of x^y shews that A must be zero, and a comparison of the coefficients of xh/ shews that B = l. [See Ex. 3, Art. 522.] 18. On expansion, the determinant = - 2a {46c - [b + cf] - (a + 6) { - 2c (a+ 5) - (6 + c) {c + a)} + (c + a){(6 + c)(a-|-6) + 26(c + a)} = 2(6-t-c)(c + a)(a + 6) + 2a(5 + c)2 + 26(c+a)2-|-2c(o + 5)2-8a6c = 2 (6 + c) (c + a) (a + 6) + 2 (62c + 6c2 + c^a + ca2 + a2j _(. a62 + 2a6c) = 2(6 + c)(c + a) (a + 6) + 2(6 + c)(c + a)(a + 6). Alternative Solution. If a + 5 = 0, so that b= -a, the determinant -2a 2a c+a c-a c + a c — a -2c = 2a {4ac + (c - a)2} - 2a (c + a)2 = ; hence the determinant is divisible by a + 6 ; similarly it is divisible by 6 + c and c + a; and therefore must be equal to k{b + c) (c + a) (a + 6). To find k, put a = 0, 6 = 1, c=l; thus 2k = = 8. 19. It is easy to shew that the determinant vanishes when a = 0, 6=0, c=0; hence it is divisible by a6c. Again, the determinant {b + cf-a^ a' b^-ic + a)" (c + a)2-62 6^ c^-ia + b)" (a + b)"- XXXIII.] DETERMINANTS. 227 Here both the first and second columns contain a + b+casa, factor ; hence the given determinant must be divisible by (a + b + c)'', and since it is of six dimensions the remaining factor must be of the form ]c (a + b + c). Thus the given determinant must be equal to kabc {a + b + c)'. Hence k must be equal to the coefficient of a*bc in the expanded d eter- minant. Now the term a*bc can only arise from the product {h + c)' (c + aY(a + b)-, and its coefficient in this product is 2 ; hence k = 2. 20. As in Art. 498, the product of the determinants and 2/3 ai^i + biVi + Vi Oa^i + hVi + '^sh a^x^ + b^^ + c^^ 03X2 + 631/2 + 0322 c b X a a hH <'2^3+M3 + ''2^3 as^a + hVs + 's' c b — c' + b' ab ac c a ab c'^ + a^ be b a ac be 62 + a2 a^ 62 'a + M'/3 + W7=0. Eliminating o, /3, y, we obtain = 0. 28. By Art. 495 the determinant can be expressed as the sum of eight determinants. The terms containing x' will be obtained from a^x aix acx or abcx' a a a aix IH hex b b b OCX bcx c^x c c c Thus the coefficient of x' is zero. The terms containing x' will be obtained from u abx acx w' b^x bcx v' bcx i.fl 1 1 1 1 1 _ 2 1 1 111 (.,2 1 1 U -2 1 1 1 1 a 0,2 1 1 It! w2 1 -2 1 1 For the constituents of the first row of the determinant-product are l + ut' + oi'^ + l; u + u' + w^ + l; w^ + w + M^ + u; I + m + u' + w^; but since u3 = l and l + u + ti,2 = 0, these reduce to 1, 1, -2, 1 respectively. Similarly for the other rows. The numerical value of the determinant on the right 1 = 27 1 1 3 -3 1 -1 2 3 -3 3 -1 1 1 -3 3 Thus the sc[uare of the given determinant is egual to - 27. 11. The determinant formed by the minors of the determinant u, h g is ic-f^ fg-ch hf-ig h b f fg-ch ca- g^ gh - af g f c hf-bg gh-af ab-K^ This second determinant is therefore the square of the first. [Art. 498.] Hence the second determinant = {abe + 2fgh - ap - 6^^ - cft^)^ But in order that the three given equations may hold, the second deter- minant must vanish; hence abc ^-ifgh ~ af^ - hg^-clfi=0. Alternative Solution. From the first two equations, the value of x is proportional to (ch—fg) (af-gh) - (bg- hf) (g''-ca), which is equal to g {abc + 2fgh - aP - bg^ - eft^). Similarly the value of y is proportional to f{abc + 2fgh - ap - bg^ - ch^) ; and the value of z is proportional to c (abc + 2fgh — af^ - bg'' - cli^). Substituting in the third equation, we have (abc + 2fgh - a/2 - bg^-ch^) {g (bg - hf) +f{af- gli) + c (/i^ - ab)}=0; that is, (abc + 2fgh - ap - bg'' - cli')'=0. 236 DETERMINANTS. 12. The solution is similar to that of the next example. [chap; 13. We may express the value of any one of the unlcnown quantities as the quotient of one determinant by another ; thus the value of tj is given by the equation 6 y= k that is, whence a** A^ c* alio (b-c) (c-a) (a -6) y = akc{k-c) (c-a){a-k); y= _k{k-c) {k-a) ~ b{b ~c){b- a) ' 14. Here 1 b 63 that is, (b-c) {c- a) {a- b) (a- d) (6 -d){c-d)u = {b-c){c~ a) {a -b){a- k) (b -k)(c-k); _{k~a){k-b){k-c) [Art. 50j] whence {d -a) (d- b) (d-c)' Examples 15 and 16 may be solved after the manner of the first solution of XXXIII. a. Ex. 24. Here however we shall give another method. 15. When d = a, the S"* row=(-l) x 2°'» row; when d = b, the 3'''' row = ( - 1) x 1" row ; when d = c, the 2"'' row = ( — 1) x 1" row. Hence the determinant vanishes in all these cases, and since it is a cubic in d it must be equal to f [a, b, c) x (a - d) (b - d) (c-d). To find the value of / (a, 6, c), put d = 0; then hence abcf{a, b, c) = f{a, b, c) = b + c-a be abc c + a — b ca abc a + b-c ab abc 1 b + c-a be 1 c + a-b ca 1 a+b-c ab and therefore f{a, b, c) = bc {{a + b-c) - (c + a-b)] + + . = 2{bc(b-c) + ca{c-a) + nb(a-b)} = -2(b-c)(c-a)(a-b). XXXIII.J DETERMINANTS. 237 16. For tlie new second column, multiply the first column by - 2, the last by - 2, and add the results to the second ; thus the determinant 6c ca ab --{a^ + b^ + e^) be ca ab = (a^ + b'' + c^} [b-c) {c - a) {a-b)f{a, b, c) ; where / [a, b, c) being of one dimension must be eq^ual to k(a + b + c). It ia easy to see that k = l. 17. Adding together all the columns we see that the determinant is divisible by a + b + c + d + e +/. Multiplying the columns by 1, -1,1, -1,1,-1 respectively, and adding the results we see that a-b + c-d + e -f is a factor of the determinant. Multiplying the columns by 1, w, m^, 1, a, w^ respectively, and adding the results, it follows that a + ab + u'^c + d + oie + a^ is a factor of the determinant. Similarly we may shew that a + u'b + b!C + d + ah + oif; a — uib + w^c — d + ue — a'f ; a — a% + ac — d + a'-e — inf are factors of the determinant. Hence the determinant is the product of these six factors and some constant, which is obviously unity. Taking these factors in pairs, it follows that the determinant is the product of the three expressions {a. + c + ef-(b + d+fY; (a + a'-c + wef-(d + a^f+abY; (» + uc + a^ef - (d + w/+ u^bf. The last of these factors = (a2-d2 + 2ce-26/)+u(e2-62^.2ac_2i/) + M2(c2-/2 + 2ac-26d) where A, B, G have the values given in the question. Similarly the second factor= 4 + oi'B + wC, and the first factor =A+B + C. Hence the determinant = {A+B + C){A + aB + w^C) {A + or'B + wC) = A^ + B^+(P-SABC, which is the expanded form of the determinant on the right side. 238 DETERMINANTS. [chap. 18. The determinant in question is 1 1 1 1 1,.. 2 3 4 5 6... 3 6 10 15 21... 4 10 20 35 56... 5 15 35 70 126... [Art. 393.] If we form a, new determinant by subtracting each row from the row immediately beneath it, we obtain a determinant in which each constituent of the first column vanishes except the first ; thus the determinant 1 2 3 4 5... 1 3 6 10 15... 1 4 10 20 35... 1 5 15 35 70... This determinant consists of n — 1 rows, and the constituents of the succes- sive rows are easily seen to be the first n - 1 terms of the figurate numbers of the 2'"', 3"', 4'", ... , m"- orders. [Art. 393.] In like manner the last determinant 13 6 10... 1 4 10 20... 1 5 15 35 ... The constituents of the successive rows of this determinant are the first n - 2 terms of the figurate numbers of the 3''', 4"", . . . , m"" orders. Proceeding in this manner, the determinant will at length reduce to I 1 n-1 1; I 1 n \ and therefore its value is unity. EXAMPLES. XXXIV. a. Pages 438-440. Here the multiplier is 3 -5; hence as in Art. 515, 11 90 1 5 '20 -10 4 20-10 -19 53 50 -15 5 31 -102" XXXIV.J MISCELLANEOUS THEOREMS. 239 2. The given expression vanishes when x = 3; hence 162-189 + 8o + i = 0; that is, 3a + 6 = 27. 3. As iu Art. 517, we have 1 1-5+9-6 -16 + 13 + 3 + 3-2 -2 -6 + 4 + 3 - 2 + 3-2 1-2+1+1 -15 + 11. Therefore the quotient is i' - 2a;'-' + a; + 1 ; and the remainder is - 15a; + 11. 4. As in Art. 517, we have 1-2-4 + 7 1-2 + 3 + 19-31 + (12 + a) -14 + 10 + 21-15 + 0+ + (a-3j. Thus the remainder is a - 3, and wiU therefore vanish when a = 3. 5. As iu Art. 517, we have 1 + 5 -7 -1 + 8 1 +5- 7- 1 + 25-35 + 90 + 8 - 5+ 40 -126- 18 + 144 + 270-378- 54 + 432 1 + 5 + 18 + 54 + 147-356+ 90 + 432. Therefore the quotient is x-^ + 5x-^ + lSx~^ + 5ix~''; and the remainder is 147a;-^ - SbBx'" + 90a;-« + 432a;-7. 6. a{b-c)^ + b{c-a)^ + c{a-b)'' = k{b-c){c-a)(a-b){a + b + c). To find k, put a = 2, 6 = 1, c = 0; thus -6/c= -6; that is, k = l. 8. It is clear that (a + b + c)^-{b + c-a)^-{c + a-b)^-{a + b-c)^ = kabc; and on putting a = 6 = c = l, we find 4 = 24. 9. a{b - c)" + b {c - a)^ + c {a- h)^ + 8a6c vanishes when 6 = - c and is there- fore divisible by b + c. Similarly it is divisible by c + a and a + b; it must therefore be equal to fc (6 + c) (c + a) (a + 6) . On putting a=l, 6 = l,c = 0, we find that A;=l. 10. This expression is equal to {b-c){c-a)(a-b){lc{a!' + b^ + c^) + l{bc + ca + ab)}. If a=2, 6 = 1, c = 0, then -14=-2(57c + 2i); andif a=l, 6= -1, c=0, then 2 = 2{2k-l); whence we find k = l, 1 = 1. 240 MISCELLANEOUS THEOREMS. [CHAP. 11. This expression vanislies-whena=0, 6 = 0, c = 0; and also when 6 +c = 0, c + a=0, a + b = 0. Thus the expression is equal to ia6c(6 + c)(c + a)({y+zy^ + 2xY{x + !'){y + z) + 2xV{x + y){z + y) + 2j/%2 [z + x){y + x)+ ix^yz {y-¥z)+ ixy^z {z + x) + ixyz^ {x + y] + ix-y^z- = Zx* {y + zy + 2Xx^y^ + 623?y^z + lOai'i/ V = Zx^{y + z)^ + 2 (2a;3t/3 + SSx^i/^ + Gx^^z^) - 2x^yH'^ = Sa;< (!/ + 2)2 + 22 (a:2/)3 - 2x^y'^z\ [Art. 522.] 22. On multiplication, we have S {ab - c^) (ac - 6^) = a%c + ah^c + afto^ _ ajs _ a^s _ aSj _ 5^3 _ gS,. _ js^ + ja^s + aS^s + ^2 ja = (6c + ca + a6)2 - (a'^ftc + aft",. + ab(? + 06^ + ac^ + a^i + 6c' + a^c + 6=c) = (6c + ca + a6)2 - (a^ + 6^ + c^) (6c + ca + a6) = (6c + ca + a6) (6c + ca + a6-a2-62-c2). 23. 06c (2a)3 - (26c)3 = a6c(Sa3 + 32a26 + 6a6c)-(26V + 32a'6=c + 6a262c2) = a6c 2a3 + 32a36=c - S6-*c' - 32a'62c =a5c2a'-26V. This last expression = a6c ((js + 63 + c3) - (6V + (?a? + a%^) = 6c(a^-6V) + a63(6c-a=) + ac3(5c-a2) = (a2 - be) \bc (a2 + 6c) - 06' - ax?) = (a2_6c)(62-ca)(c2-o6). 24. Let 6-c=s, c-a = y, a-b=z, then we have to shew that 23^{y -z) = when a; + y + 2=0. Now ifi{y -z) + y^{z- x)+z^{x-y) = k{y~z]{z-x){x-y){x + y + z); which vanishes because of the zero factor x+y+z. 25. The solution is similar to that of the next Example. H. A. K. 16 242 MISCELLANEOUS THEOREMS. [CHAP. 26. From a formula given in Art. 523, we have =^(x+y + z){{y-!:)'' + {z-x)^+{x-yY} =^(a + ft + c){4(6-c)2+4(c-a)2 + 4(a-6)=} =4(a3 + 63 + c3-3a6c). 27. Let a: = s-o, y=s-b, z=s-e. By Art. 523, xS+yi + z^-3xyz=~(x + y + z){{y-zf + {z-x)^+{x-tjY} = |(3s-a-6-c){(6-c)2+(c.-a)H(a-6n = |(a + 6 + c){(6-c)2+(c-o)2 + (a-6)y =a' + 6' + c'-3a6c. 28. The common denominator is (6 - c) (c - a) (a - 6) (s -a){x- 6) (a; - c), and the numerator = - Sa (6 - c) (x - ft) {x - c) = -a:2Sa(6-c) + aSo(6-c)(S + c)-ffl&c2(J-c) =a;Sa(62-c2) = (6 - c) (c - a) (a - 6) x. Note. It is easy to prove the converse result by resolving 7 -, :—, r- {x-a)[x-b)[x-c) into partial fractions. 29. The common denominator is (6 -c){c- a) (0-6), and the numerator = - 2a= {b-c) + '2{V + c^) (6 - c) = -Sa''(6-c)+S{(63-c3)-6c(6-c)} = -Sa=(6-c)-S6c(6-c) = (6 - c) (c - a) (a - 6) + (6 - c) (c - a) (a - 6) = 2(6-c)(c-a)(a-6). 30. The common denominator is (6 - c) (c - a) (a - 6) (x + a) (x + 6) (x + c) ; the numerator = - S (a +p) {a + j) (6 - c) (x + &) (x + c) = -2(rt+j))(a + 3){x2(6-c) + x(i2_c2) + 6c(6-c)}. XXXIV.J MISCELLANEOUS THEOREMS. 243 The coefficient otx'=- Sa^ (6 - c) - (p + q) Sa (6 - c) -pjS (6 - c) = {b-c){c-a){a~b). The coefficient of x= -j:,a''{b'-c')-{p + q)I,a{b^-c^)-pq-2(b^-c^) ^{p + q){b-c)[c-a)(a-b). The term independent of x = -o6cSa(6-c) -abc{p + q)S{b-c)-pqXbc{b-c) =:pq (6 - c) (c - a) (a-b). Thus the numerator=(6-c)(c-o) (a-6){a;2+(pH-5)a;+pg}. Note. The converse of this result is easily proved by Partial Fractions. 31. The numerator is - S6cd(6 - c) (6 - d) (c - d). This expression is of six dimensions, and vanishes when 6=c, or c = a, or a=b, or a=d, or 6 = d, or c = d; hence it must be equal to k{b-c){c-a){a-b)(a-d)(b- d) [c-d). A comparison of the coefficients of bVd shews that k= -1; hence the numerator becomes ~(b-e){c~a){a-b){a-d){b~ d) {c-d). 32. The numerator= -2a<(6-c)(6-d)(c-d). This expression consists of four terms and vanishes when 6=c, or c = a, or a = 6, or . Substituting for x in* succession the values 1, u, u^, the terms involving abc destroy each other, and by adding the results the other terms are zero, since l + a) + 0)^=0. 6. This theorem is involved in that of Art. 525. 7. By writing -i for 6, ~y for y and putting c = 0, 2=0, the theorem follows from that of Example 6. Or we may prove it directly as follows : (a^ + ab + b'')(x^+xy+y^) = {a-ab)(a-by'b)(x-t^)(x-a'y). Now {a-ab)(x-ahi) = ax + by-oibx + {l + w)ay = ax + by + ay-u(bx-ay); and (a-u'^b)(x-uy) = ax + by-a%x + (l + w^)ay = ax + by + ay-u'^{bx-ay). Thus the product =(A-wB)(A-a'B) = A^+AB+B% where A = ax + by + ay, B=bx-ay. 8. LetX=a'' + 2bc, Y=b' + 2ca, Z=c^ + 2ab; then X+Y+Z={a + b + c)^ X+u>Y+a''Z=(a + u^b + ac)^, X + ui^Y+aZ={a + ub + oi^c)\ XXXIV.] MISCELLANEOUS THEOREMS. 245 9. Let X=a?-bc, Y=V-ca, Z=c^-db, then X+Y+Z = {a + ub + u?c)(a + b^{{a-bf + iab} = 8(o-6)« + 72a6(a-6)< + 162a262(a-6)2 = 2(o-6)2{2(a-6)2 + 9a6}2 = 2(a-6)2(2a2 + 5a6 + 262)2 = 2 (a -6)2 (2a + 6)2 (a + 26)2 = 2(a-6)2(a-c)2(6-c)2, since a+5=-c. The theorem may also be deduced from Art. 574, Ex. 2. 20. Put 6 — c = o, c-a=j3, a-6=7, so that a + /3 + 7 = 0; then we have to shew that o^ + p8 + 78 - 3a2;8V = 2 f^L±^±f.\ ' . This is easily proved from the Note preceding the solution of Ex. 16, for a<'+p« + 7'==3r2-2g5, o/37=r, 0^+^^ + ^^= -2q. 21. Proceeding as in Ex. 20, we have to prove that a'' + 0' + y''=7a^y(~^^^j . [Seethe Note preceding Ex. 16.] 22. Suppose that a=0, in which case c= -5; then the left-hand side becomes 4:b^{y-z)^-eb^{y-z){x^ + y^ + z^)-ib^{y-z)(z-x){x-y) = 2V{y-z){2(y~z)^-3{x'' + y'^ + z'') + 2{x~y){x-z)} = 2b^y-z){-x''-y^-z''-2xy- 2xz - 2yz} = -2b^{y-z){x + y + z)^ = 0, since x + y + z = 0. XXXIV. J MISCELLANEOUS THEOREMS. 247 Hence the left-hand side vanishes when a=0; and similarly it vanishes when 6 = 0, c=0, x = 0, y = 0, 2 = 0, and thus may be put equal to kabcxyz. To find k, put a=l, 6=1, c= -2, x = l, y = l, z= -2; thus 4ft=4x63-3x6x6x6=63; whence ft = 54. 23. Assume {l + ax){l + bx){l + cx)(l + dx) = l + qx' + rx^ + sx*, where g = Sa6, r=Sa6c, s=a6c(J. By proceeding as in Art. 526, we have Sa2 = - 2j, SflS = 3r, Sa" = - 5gr j whence _ = _.__. 24. With the notation of the preceding Example, we have (SaS)2=9r2 = 9(Sa6c)2. Since d= - (o + 6 + c), we have bcd + eda+dab + abc= -(a+h + c){bc + ea+ab) + abc— - (6 + c) (c + a) (a + 6) . .-. (bed + cda + dab + abc)^={b + cy{c + af (a + bf. But (c + a)(a + b) = bc + a(a + b + c) = bc-ad; and similarly (a + b)(b + c) = ca-bd, and (6 + c) (c + a) = a6 - cd. 25. We have 8 (s - 6) (s - c) (o-^ - a?) = (a-l + c)(a + b-c){b'^+c'-a?) - {a? _ 62 - c2 + 26c) (62 + c2 - a^) = 26c (b^ + + c'z+d' = 0; 23. Divide by j/' and put -=z, then we have a bz^ + cz + d az + b cz + d az^ + bz + c a' b'z' + c'z + d" a'z + V c'z + d" a'z^ + b'z+c' d'' Multiply up and eliminate z^ and z from the three equations so obtained. EXAMPLES. XXXV. a. Page 456. 4. Corresponding to the first pair of roots there will be a quadratic factor X? - 2ax + (a^ - b") ; and corresponding to the second pair a quadratic factor x^ + 2ax + {a^ - b^) . Thus the required equation is {x2 + (a2-J5)}2-4(i2a;2=0. 5. Corresponding to the two given roots there is a factor x^ - 8x+ 7. By writing the equation in the form x-{x^-8x + 7)-8x(x^-8x + 7) + 5{3^-Sx + 7)=0, we see that the other two roots are obtained from the quadratic equation x^-8x+S=0. 6. Let the roots be a, -a, b; then the sum of the roots = 6 = - 4 ; 9 also the sum of the products two at a time= - o'= - - . 254 THEORY OF EQUATIONS. [CHAP. 7. Let a, u,, bhe the roots ; then OQ 3 2a+6=-5; a^ + 2ab = ^; a%=-^. Eliminating b from the first two equations, we get 120^ + 40a - 23 = ; 1 ^'S 8 whence a== or - ^ ; and from the first equation we find 6= - 6 or --. 23 8 It will be found on trial that a=-^. b= --^ do not satisfy the third o ' o 3 11 equation a%= --; henoe the roots are ;; , ^ , - 6. 8. Let ~ , a, ar be the roots; then a? = %, and a(r+l+-| = — ; 1 2 ^ ''^ whence a=2, r=3 or - ; thus the roots are - , 2, 6. A o 9. Let 3a, 4a, b be the roots; then 7a + 6=i, 7a6 + 12a==-ll, a^b = l. From the first two, by eliminating 6, we have 74a°-7a-22 = 0; whence 1 22 1 3 a= -^ or — . Taking a= --, we get 6 = 4; thus the roots are --, -2, 4. It will be found that the other value of a is inadmissible. 10. Let 2a, a, i be the roots ; then 3a + 6=-j|, 2a2 + 3a6=|, 2a%=^. By eliminating 6 from the first two of these, we get SGa^ + 46a + 3 = 0; 3 1 1 whence <»=-t or -rj. The first of these values gives 5=5 1 the other being inadmissible. 11. Let a, -a, b, c be the roots ; then 6 + c=i. (b + c)a^=l, -a='6c=|. 3 1 3 1 Thuso=±v/3, 6c=-^, 6 + c = t; whence 6=^, c= — . 12. Let -, a, ar be the roots; then a'= -— =, a ( — hl+ri = — ,. r 2,1 \r j 54 Thus a= --, and 12r2 + 25r + 12=0; whence r= -^ or -|. o 4 XXXV.J THEORY OF EQUATIONS. 255 13. Let a-d, u,, a + d be the roots; then, as in Art. 541, we have a=-rr, Za^-d^=^r^; whence d=±-. 2 16 4 14. Let a, 6, c, d be the roots, and suppose that C(J=2 ; then ,29 ^ J 1 , J J 40 a + o + c + a=-=-. ab + (ic + ad + oc + oa + cd= ,. . 6 o 7 a6c + acd + a6(J + 6cd = ;; , o6cd=-2. D Thus ab= -1. By substituting a6= -1 and cd = 2, we have 7 29 -c + 2a-d + 26=- , c + a + «Z+6 = -5- ; D D whence by addition, a + 6 = 2; and since a6= -1, we easily obtain 1±;^2 for two of the roots. We now have c + d= ^, cd = 2; whencec = 5, d= = . 15. Let a-3d, a-d, a + i, a + 3d be the roots; then 4a = 2, and a = ^ . Also (a3-9d2)(a2-d2)=40; hence (l-36(i2)(l -4(P) = 640; that is, 144d* - 40^2 - 639 = ; or (id^ - 9) (36(i2 + 71) = ; 3 thus d= ± ^ ; and the roots are - 4, - 1, 2, 5. 16. Denote the roots hy ^ , - , ar, wfi ; then the product of the roots =a*= ^r=- = -^ ; whence a?=^. The sum of the products of the roots two at a time =«^(i + ,4 + 2 + '-» + r^) = ^S thu. (,.+ iy+(,.+ i)=^7. 1 13 3 whence r^+ -5 = -^ » and therefore r^ = « • Q Q 4 8 Thus aV2= - X - = 4, or ar= 2 ; and therefore the roots are 8, 2, ^ , - , 256 THEORY OF EQUATIONS. [CHAP. 17. Let a, h, — =- be the roots; then 3, ,, 81 ab(a + h) 10 whence, a + 6=-3, ab=—; and therefore a=-^, 6=--. 18. (1) Here we have a + 'b+c=p, db + 'bc + ca = q, abc — r. 1 _ S {a%') ._ (ab + 6e + eg)' - 2abc (a + b + c) _ q''- 2rp ■■ a" - aVc' ~ a'b'c' ~ 7- m V 1 __ a"- + b^ + c' _ p'-2q ^' d^b^ d'h-c^ r" ' 19. (1) Here a + b + c=0, ab + bc + ca=q, abc= -r. .: S(b-c)2=2(a2 + 6=+c2)-2(6c + ca + a6) = 2(a + 6 + c)2-6(6c + ca + a6) = -6?. (2) Since a + b + c = 0, we have ^-)-K-9=-^ 20. (1) Here we have Sa=0, ^ab = q, 'Zabc=-r, abcd=s. .: 2a^= (Sa)2 - 2Sa6= - 2q. (2) Again, Sa3=32oZ)c= -3r. [See XXXIV. b. Ex. 23.] 21. Here Sa=0, Sa6 = g, o&c= -r. Multiply the equation through by X, then substitute a, i, c for » successively and add the results; thus we obtain 2ia* + q'Zd' + rXa=0; :. 2a*= -2Sa2= -2 {(2a)2_22a6} = 222. EXAMPLES. XXXV. b. Page 460. Examples 1 — 12 do not require full solution, as they all depend on Arts. 548 — 545, and the method of procedure is explained in Art. 545. As further illustrations the following solutions will be sufficient. XXXV.] THEORY OF EQUATIONS. 257 1 + / — ~S 1 — / — ^ 1, Corresponding to the two roots ^ , ^ we have the quadratic factor x^-x + 1. The equation is now easily put in the form (Zx' - 7x - 6) (a" - a; + 1) = 0. Thus the other roots are obtained from 3x^-7x-e = 0. 5, Here four of the roots are ±^3, lJz2iJ - 1. Corresponding to these pairs of roots we have the factors a;^-3 and x'-2x + 5. Aloo the equation may be written (x + 1) (x^ - 3) (x^ - 2a; + 5) = ; hence the remaining root is - 1. 6. The equation required has the following pairs of roots : +v/3 + V^, +V3-V^; -n/3+s/^, -^/3-V^. Corresponding to these we have the quadratic factors a;2_2^3.x + 5 and a;2 + 2^3.x + 5. Thus the equation is (x^+2^3.x + 5)(x^-2sj3.x + 5) = 0, or s* - 2a;2 + 25 = 0. 10. Here we have the quadratic factors x^ - 48 and x'^ - lOx + 29 cor- responding to the two pairs of roots ; hence the equation is (x2-48)(x2-10x + 29) = 0. 12. The equation whose roots are ^2 + ,^3 ± J -X is (x-V2-V3)'' + l = 0, or x= + 6-2J2.x-2J3-^-2x/6 = 0. Similarly the equation whose roots are sJ2-J3J=J -1 is x2 + 6-2^2.x + 2^3.x + 2V6 = 0; these two equations are equivalent to (x2 + 6-2^2.xf-(2V3.x + 2V6)2=0, or x«+8x2 + 12-4;^2 . x3=0. Hence the equation whose roots are -iJ2^^3±^ -1 is x* + 8x2 + 12 + 4;^2.x' = 0; thus the required equation is (xH8x2 + 12)2-(4x/2.x3)2=0, or x^ - 16x8 + 88xH 192x2 + 144 = 0. 13. Denote the equation by /(x) = 0; then in /(x) there is one change of sign, so that there cannot be more than one positive root. Again, /(- x) has only one change of sign ; therefore there cannot be more than one negative root. Hence there must be at least two imaginary roots. [By Art. 5S4, we know that the equation has one positive and one negative root.] 14. Here f{x) has three changes of sign, and /(-x) has no change of sign. Therefore the equation has no negative roots and at most three positive roots. Hence it has at least four imaginary roots since it is of the seventh degree. H. A. K, 17 238 THEORY OF EQUATIONS. [CHAP. 15. Here f(x)=x^''-ix<' + x*-2x-3, and f{-x) = x^'>~4x^ + x* + 2x-3; thus there are three changes of sign in f{x), and three changes of sign in f{-x); hence there cannot be more than three positive roots nor more than three negative roots; henoe at least four of the roots must be imaginary. [By Art. 554, we see that th^e equation has one positive root and also one negative root.] 16. Since f{x) has two changes of sign, the equation has at most two positive roots. And since /( -x) has only one change of sign, there cannot be more than one negative root. Therefore it must have at least six imaginary roots. 17. (1) Let a, i, -b be the roots; then a=p, -V=q, -aW=r; by eliminating a, b, we obtain jig=r, which is the required relation. (2) Let =-, u, ak be the roots; then 'a a^ a^=r, ■^ + a+alc=p, -j- + a^ + a,''lc=q; K K P 1 thus - — - , or p'a' = 2" ; that is, p^r = g'. 18. Let a-dd, a-d, a + d, a + 3d be the roots. Then by Art. 539 we have after easy reduction the relations ia=-p, 6a^~10d' = q, ia^-iOad'^ -r. From the last two of these equations we have, on eliminating d, 12a^ - 2aq=ia^ + r, or 8a^-2aq=T. Multiply by 8, transpose, and put 4a=-p; thus we ohtain. p^ - ipq + 8r=0. In the second case assume for the roots, a, ak, ah'', ak^; then we have a(l + ft + 7c2 + i;3)=-j), a'k'>(l + k + k'' + k^)=-r, aVi^=s; whence it is easily seen that ^^s=r^. 19. Put l-x=y, then we have (l-?/)"-l = 0. Expand and divide by y ; thus yn-i_nyn-^+ n (n-l) j/"-3- ... + ( - l)''-ira=0. If 2/ii 2/2> 2/31 ■••> 2/n-i denote the roots of this equation, we have 2/iJ/2!/3--2/n-i=n; thatis, (l-a)(l-j3)(l-7) = n. 20. Here Xa=p, 2ab = q, abc=r. .: 2a562^(2a6)2-2a6c2a = 2'-2rp, XXXV.] THEORY OF EQUATIONS. 259 21. Here {i + c){c + a){a + b) = {ab + bc + ca)(a+b + c)-aT)c=pq-r. 2- ^-^^^9=^"^ =-{ab^ + ac' + bc^ + ba? + ca' + c&2) = - { (a + 6 + c) {a6 + 6c + ca) - 3a?)c} =J(M-3r). 23. Here 'Ea%-a'b + a?c + bH + b''a + c^a + c^b —Pi ~ 3''> ^s in Example 22. 24. Here we have S(i= -p, Sa6 = g, So6c=-r, a5cd=s. Now Sa%c = a2(6c + 6d + c(i) + + + = a (abc + abd + acd) + + + =a(-r-bcd) + + + = -r{a + b + B + d)- 4.abcd =pr - is. 25. Substitute a, b, e, d for x suooessively and add the results; thus we obtain 2a* i-p'Sa^ + qZa!' + rSa + 4s = 0. Now 2a= -p, and Sa^=p^-2g. Also 2a3=(2a)3-32a26_6Sa6c [Art. 522.] = -p^-3(3r-pq) + 6r. [Art. 542, Ex. 2.] .". 2a*=p* + 3p (3r -pq)-6pr~q (p^ _ 2g) +pr - is =p* - ip'q + 2q^ + ipr - is. EXAMPLES. XXXV. c. Pages 470, 471. 1, Proceeding as in Art. 549, we have 10 39 76 65 6 15 16 1 1 2 7 1 -12 - 2 |15 ll - 6 .■./(^- -i) = x*- -6^3 + 15a;' -121 + 1, 17—2 260 THEORY OF EQUATIONS. [CHAP. 2. We have 1-12 17-9 7 1 - 9 -10 - 39 I -110 1 _ 6 - 28 I - 123 I - 3 I -37 II .-. /(a; + 3) = a^ - 37x2 - 123a; - 110. We have 2 -13 10 -19 2 2 - 11 -11 -20 2 4 - 7| - 8 2 6| - 1 2 |8 .•.f{x + l) = 2xH8x^-x^- -8a; -20. We have 16 72 64 -129 12 24 -32 1 - 1 8 - 8| 4| -24 1| .-. /(x-4)=a;«-24a;2-l. 5. By Art. 548, we have /(a; + ;>)-/(x-70 = 2|ft/'(a;) + |/"'{x) + ^V''(^) + ^V''"W}- Now/'(a;) = 8aa;'' + 56x* + c; f"'(x) = 8.7 .6ax^ + 5 .i.Shx^; and therefore • ■^^=56ax5 + 106x2; ii also '^^=56ax3 + 6; •^^ = 8ax. [5 17_ .■.f(x + h)-f(x-h) = 2 {7{ (8ax' + 5&X* + c) + h^ (56ax5 + 106a:2) + h^ (56axS+ 6) + W . Sax}, which easily reduces to the form given in the answer. 6. Here/(0) = 6, and/(-l)=-22; thus /(O) and /(-I) have different signs, and therefore there is a root between and - 1. 7. Here /(2) = 16- 40 + 12+ 70-70= -12; /(3) = 81 -135 + 27 + 105 -70 = 8; -'• f[^) = ^ ^^^ ^ "^0°* between 2 and 3. Again, /(- 2) is negative, and /(- 3) is positive; therefore /(x)=0 has a root between - 2 and - 3. Examples 8 and 9 may be solved in the same way. XXXV.] THEORY OF EQUATIONS. 261 10. Here f(x)= x*- 9a;2 + 4a; + 12, and f't,x) = ix^-18x + i. The H.C.F. of these two expressions will be found to be x-2. Thus (x - 2y is a factor of f(x). Now f{x) = {x-2)^{x^ + ^x+3) = {x-2)^(x + 3)[x + l); thus the roots are 2, 2, - 3, - 1. 11. Proceeding as in Ex. 10, we find that the H.C.P. ot f{x) and f'(x) is x^-^x + l; hence (x - Vf is a factor of / (a;). Now /(a;) = (a!-l)'(a!-3); thus the roots are 1, 1, 1, 3. 12. Here S(x)= x»-13a;*+ 67x3- 171a;2 + 216a;- 108, and /'(a;) = 5x*-52a^ + 201a:='-342a;+216. .-. 2/ (x) +/' (x) = X (2x4 - 21x3 + 82x2 - 141x + 90) = x0 (x) say. .-. 2/' (x) - 5^ (x) = x» - 8x2 + 21x - 18 ; and since this expression divides ^(x), it is theH. C.F. of /(x) and/'(x). Now !c'-8x2 + 21x-18=(x-3)2(x-2); and /(x) = (x-3)3(x-2)2=0; and therefore the roots are 3, 3, 3, 2, 2. 13. Here /(x) = a;i>-x3 + 4x2-3x + 2, and /'(x) = 5x4 -3x2 + 8a! -3. The H.C.F. of these expressions \%a?-x-^\. Hence / (x) = (x^ - x + 1)^ (x + 2) , and the roots are 1±7^ l±s/^3 -2. 2 ■ 2 ■ 14. Here it will be found that /(x) = (2x - \f (x + 2) ; thus the roots are k > 5 > 5 > ~ 2' 15. Here it will be found that /(x) = (x-l)3(x3-3x-2) = (x-l)3(x + l)2(x-2). Thus the roots are 1, 1, 1, - 1, - 1, 2. 16. Here /(x) = (x=-3)2(x''-2x + 2). Therefore the roots are ±^^3, ±^3, 1±^^. 17. Here /(x) = (x-a)2{x2-t-(a-6)x-(i5}. Therefore the roots are a, a, 6, - a. 262 THEORY OF EQUATIONS. [CHAP. 18. Denote the two equations by f{x) = and F(x)=Q. Then it will be found that the H.C.P. of f{x) and ^(3;) is Sx^-S. Also f{x) = {2x^-3){x^-x + 2); and F{x) = {2x^-S)(2x^-x + d). Thus the roots are =>= * / g . — ^| ! ^'"'i \/ 2 ' — 4 " Example 19 may be solved in the same way. 20. If f{x) = has equal roots, f{x)=0, f'(x) = have a common root. Thus x^-px^ + rz^O (1) and na;»-i-2^a; = (2) have a common root, and the required condition will be obtained by elimi- nating X between them. 2p From (2), we have x"-^= — . n Multiply (1) by n and (2) by x; then by subtraction, p(n-2)x^=nr, that is, x^ = - Xn-2)- f nr ] "-1 _ /2j)Y 21. If /(x) = has three equal roots, /(s) and fix) must have a com- mon quadratic factor, and /'(a;)=0 must have too equal roots. Now /' (x) = 2a; {2a;'i -t- 5), so that one root of/'(a;) = is zero, and the other two are the roots of 2a;''' -f- 3 = which are equal in magnitude but not in sign. Thus f(x) = cannot have three equal roots. 22. We may write the two equations in the following forms : x^+~x + l = Q (1), a and (x-l){x^-x + l) = (2). Now in (2) we have one real root and two imaginary roots. Therefore by Art. 543 the two equations may have one common real root, or tioo common imaginary roots. In the first case a; = l satisfies (1), and thus 6 = - 2a. In the second case x^ -i- - a; -I- 1 must be identical with the quad- a ratic factor x^ - a; -t- 1, and thus 6 = - a. 23. Here we have f(x)=x'^+nx'<'-^ + n(n-l)x'^-^+ ... + \n =0, and /'(a;)= nx"-! -Hn(n-l)x''-2 -(-... -H In =0. Now if f{x) has a pair of equal roots, f(x) and /' (x) must have a factor of the form x-a. Therefore also f[x) -f (x) must have a factor of this form. But /(x)-/'(x)=x», and it follows that /(x) = cannot have equal roots. XXXV.] THEORY OF EQTJATIOXS. 263 24u Here f'(x)=5ii!*-Wa^x + h* (1), and /"(x) = 20xs-20as (2). If /(x)=0 has three equal roots f'{x) and /""(.r) must have a conimou linear factor; from (2) it is evident that this factor must hex -a. Thus x=a must satisfy the given equation. On substituting for x we get the required relation. 25. Here /'W= ii^ + Sax^+2bx+c, f"{x)=12x^ + Gax + 2b; and if /(x)=0 has three equal roots, f'[x) and f"{x) must have a common factor. Hence 4i? + Sax' + 2ix + c=0 (1) and 12x» + 6aj;+26=0 (:) must have a common root. Multiply (1) by 3 and (2) by x; thus by subtraction 3ar= + -16x + 3c=0 (3). Eliminating r' between (2) and (3), we get (Ga^ - 1G&) x = 12c - 2a6 ; whence x = -r^s r > which is the common root. ia' - bo 26. Here x^+qx^+rx*+t=0 (1) and 5x^+3qx*+2rx=0 (2) must have a common root. Multiply (2) by x and (1) by 5 and subtract; thus 2jxS+3rx= + 5f = (3). Multiply (2) by g and divide by x ; thus 5qx^ + Bq-x + 2rq=0 (4). Eliminate gx* between (3) and (i) ; thus we find x is one of the roots of 15rx^'-62=x + 25f-4}r=0. 27. By the method of Art. 563 we have /(i) x-a^x-6 x-c ~ \x X- x^ '"J =- + '^, + '^. + ... + '^- + ... . X X- x" X- 264 THEORY dF EQUATIONS, [chap. Now f{x)=x^-x-l and /' (a;) = 3x^ - 1. 1 3 + 0-1 O + B + 3 1 + + 1 + 2+2 + 3 + 3 + 2 + 2 + 5+5 3+0+2+3+2+5+5+ Thus s„ = 5. 28. Proceeding as in the last Example, we have to find the coeffioienta of -^ and — in the quotient of in? - Sx' - lix + 1 by £c* - a^ - Ix' + a; + 6. 1 4-3-14+ 1 1 4 + 28- 4- 24 7 1+ 7- 1- 6 1 15 + 105- 15- 90 6 19 + 133- 19 99 + 693 . 211 + ... 4 + 1 + 15 + 19+ 99 + 112 + 795 + .., Thus 84=99, 85 = 795. EXAMPLES. XXXV. d. Page 487. 1, Put x = - and multiply each term by j'; thus 1 a" By putting g = 6 all the terms become integral, and we obtain 2/'-24!/2 + 9j/-24 = 0. Example 2 may be solved in the same way. Examples 3 and 4 are reciprocal equations which present no difficulty; they may be solved like the Example in Art. 133. 5. Here a: = l is evidently a root. On removing the factor x-1, we have x*-4:x'' + 5x'-ix + l=0; ,.(^,.+ l^)_4(. + ^)+5 = 0. , (, + iy_4(. + l)+3 = 0; XXXV.] THEORY OF EQUATIONS. 265 whence ii; + - =3 or 1. By solving these two quadratics we obtain ''-^' 2 • 2 • 6. Divide all through by a? and rearrange ; thus or 4 (x + 'X -2i(x + -Y + i5 (a; + ~\ -25 = 0. By inspection a; + - = 1 satisfies the equation. On removing the factor corresponding to this root we have the equation 4 (^xt-V- 20^x4-^^ + 25 = 0. 1 5 5 The roots of this equation in ii; + - are -, -. Thus finally we have to a; 2 2 solve the quadratics 1 , 15 15 x + -=:l, a; + - = -, x + - = -. X X 2 X 2 7. Put x=-, then the resulting equation 32j/' - 48j/2 + 22i/ - 3 = has its roots in A. P., and may be solved like Example 1 in Ait. 641. Example 8 may be treated similarly. 9. The equation 6^'- J/'' +01/ -1 = has its roots in A.P. Let them be denoted by a - d, 14, a + d ; then3a = j-; thatis,a==-. Thus the mean root of the original equation is 35. 10. The equation y* + 2y^-21y^- 22y + 40 = has its roots in A. P. Assume a-Sd, a-d, a + d, a + 3d for the roots, and proceed as in Ex. 15. XXXV. a. 11. Here, since the sum of the roots of the equation is 6, we must decrease each root by 2. We have therefore to substitute x + 2 for x, which is effected by Horner's process using x - 2 as divisor. 1 -6 10 -8 1-4 2 I 1 1 -2 I- 2 1 I 1 Thus the transformed equation is x' - 2x + 1 = 0. 266 THEORY OF EQUATIONS. [chap. 12, Here we have to increase each root by 1; therefore lising a + l as divisor in Horner's process, we have Thus the transformed equation is x^ - 4a;' + 1 = 0. 13. Here we have to increase each root by 1. Therefore using a; + l as divisor in Horner's process, we have 5 3 1 1 -1 4 -1 2 -1 3 -4 6 1-7 2 -6 |12 1 1-7 |0 Thus the transformed equation is x^ - 1i? + 12a:' - 7a; = 0. 14. Here we have to decrease each root by 2. Therefore using a; - 2 as divisor in Horner's process, we have -12 3 - 17 300 -10 -20 - 40 - 77 -171 1 - 42 - 8 -36 -112 -301 -773 - 6 -48 -208 -717 - 4 -56 -320 - 2| -60 1 Thus the transformed equation is a:8-60a;*-320a;' -717a;--773a;-42=0. 15. Here we have to use x + ^ as divisor. 1 3 " 4 4 3 2 15 2 4 13 3 2 9 2 '9 13 15 Thus the transformed equation is a;'- -x' + — a; = 0. Examples 16 and 17 may be solved in the same way. XXXV.] THEOET OF EQUATIONS. 267 18. Put y=x\ so that x=ijy ; then after transposing we have y^ + 2y + l = {y + l)^y; whence y*+iy^ + 6y'^ + 4^y + l=y{y'' + 2y + l). 19. Put y=x^, so that x=^y; 2 2 then y + 3y^ + 2=0; or {y + 2)' = {-3y^f; which reduces to y^ + 33y^ + 12y + 8—0. k 20. When x = a in the given equation, y=-hi the transformed equa- k tion. Thus the transformed equation will be obtained by substituting y = - , or s = - in the given equation. y 21. If x=a, then y = Wc^= — ^ =-2> ^2 We have therefore to substitute ;!;-=— in xlx^ + q)= -r. y Hence -,— ( - + 9 )= -»"; •Jy \y J that 13, - T + -^- +i-]=i^< yW y J or y^-qh/^ — 2qr^i/ — r^=0. b + c a 1 22. If x=a, then !/ = — ^ = - — = — ; thus we have only to substitute s= — in the given equation. «« TP ii. abc + 1 1-r 23. If a; =o, then 2/ = —^— = ^-; l-r thus we have to substitute x= in the given equation. y 24. If 1=0, then y=a{h + c)=a{-a)=-a^=-xK We have now to substitute x= J — y in the given equation. Thus(V^)(-2^ + 9)=-'-; that is, y^-2qy-^+qh) + r^=0. 25. Here as in Ex. 19 we have only to put x=yy. 1 Thus j/ + gj/" + r=0, or {y + Tf= -qhj; that is, y^ + 3ry^ + {3r' + q^)y + T^=(i. 268 THEORY OF EQUATIONS. [CHAP. 26. If. = a,then, = *!±^=M!-2=?^-2=-l'-2. ' " be be be r Therefore the transformed equation will be obtained by putting y= _(^!+2r) ^ ^^ ^^ -r(2+y). Now x' + r= -ga;; hence 2a; + r=7'(2 + 2/), or ga;=r(2/ + l); and therefore the new equation is {(!/ + l)r}3=-2V(2 + 2/), or r^j/S + 3,^2,2 + (3^2 + g3)^+y (,.2+ 2gS) = o. 27. Let y=x', so that a; = 4'!/. 1 1 From the given equation we have y-\-ah= -y^(ay^+b). Cube each side; thus 1 1 (2/ + a6)3= - 2/ {aSj/ + 63 + 3a6i/3(a2/3 + 6)} ; .-. y^ + d?b'^ + ?,aby[y-^ab)= -y{a^y -yb^ -Zab[y\ ab)\; .-. 2/5 + a3j,2 + b^y + aS j3 _ 0. 28. The sum of the roots =c = 5; hence one of the roots is 5. The equation may now be written (a;-5)(a;<-5a;2 + 4) = 0, or (x-5)(x2-4)(a;2_l)=0. Thus the roots are ±2, ±1, 5. 29. Write - for s ; then the equation has its roots in A. P. Denote the roots by a - d, a, a + i ; then 3a= — -\ that is, a= — ? . {•2 g" 3p Now o (ai* - d^) = _ - ; whence 3 f^ - dA =1, or d= Also %a?-d? = ^-^. r Thus f^(?!_l\=L^ XXXV.] THEORY OF EQUATIONS. 269 EXAMPLES. XXXV. e. Pages 488, 489. As usual, we shall denote the imaginary cube roots of 1 by w and ur^, so that2(j=-l + ^/~3, and2«2=-l-^^; also l + w + w2=0. 1. Putting x=y+z, we find 3yz - 18 = 0, or yV = 216. Also y' + z^ =216; thus 2/8=27, s^=S. Thus the real root is 3 + 2 or 5; and the imaginary roots are and 3o)^ + 2m = i.)^-2= „^ . 2. Here %z + 72 = 0, that is y'z^ = ( - 24)3 = - 123 . 23. Also 3/3 + «3 = 1720 = 123 - 23 ; whence ^3 = 123, and 23 = - 2'. Thus the real root is 12-2 or 10; and one of the imaginary roots is 12u-2u2=12M + 2(l + w) = 2 + 7(-l + ^/^) = -5 + 7^/^. The other root is got by changing the sign of ^/ - 3. 3. Here 3yz= - 63, or yh^= - 213= _ 73 . 33. Also 2/3+23=316 ; whence y'''=7^ and 23= -33. Thus the real root is 7 - 3 = 4 ; and one of the imaginary roots is 4. Here 3yz=-21, or y^z^=-l^. Also 2/3 + 23= -342; thus y=-7, z = l. The real root is -7 + l=-6; and one of the imaginary roots is -7a) + (.)«= -l-8u=3-4»y^. 5. Let x=-; then j3-9{ + 28 = 0. Putting «=y + 2, we have 3i/2=9, or 2/323=27. Also 2/3 +23= - 28 ; whence y^= - 27, and 23= - 1. Thus the real value of f is - 3 - 1 or - 4 ; and one of the imaginary values is -3(ij-a)^=l-2a=2- ^ -3. Thus the values of x are --7 , and ;^= or '^ ^ 4 2±V-3 7 6. This equation maybe written (x - 5)3 - 108 (x- 5) +432 = 0; that is, js - 108t + 432, where t=x-5. Putting t=2/ + z, we have 3yz = 108, or y^z^=3&^=6^. 63. Also ^3+^3— _ 432; \7l1ence ^3= _ 63 and z3= - 63. Thus the real value of « is -6-6 or -12; and the other roots are - 6(1) - 6w^ and - 60;^ - 6cd, which are both equal to 6. Thus the values of x are - 7, 11, 11. 270 THEORY OF EQUATIONS. [CHAP. 7. Multiply the given equation by 4, then 8a;3 + 12x2 + 12a; + 4=0, or (2a; + l)3 + 3(2a; + l)=0; ■whence 2x4-1 = 0, and (2x + l)2 + 3 = 0. 8. Here 3yz= - 12, or yh^= - 4'. Also y^ + z'^ = 12; whence ?/'=16, z'=-4. Thus the real root is 4/16 -^4 = 2^/2- ^4. Examples 9 — 17 may be solved by the methods given in Arts. 582, 583; but usually shorter solutions may be easily found. 9. Here a;*=3x'' + 42a;-|-40; on adding 6x^ + 9 to each side, we have a;^ + 6a;2 + 9 = 9x2 + 42x-)-49; that is, x^ +3 = ± (3x + 7) ; thus x2-3x-4 = 0, and x2 + 3x + 10 = 0. 10. Here x*=10x= + 20x + 16, and therefore x*-6x^ + 9 = 4x2 + 20x + 25; that is, x2-3=±(2x + 5). Thus x2-2x- 8=0, audx2 + 2x + 2 = 0. 11. Here x^ + 9x2-10 + 8x(x2- 1) = 0; that is, (x2-l){x2 + 10) + 8x(x2-l)=:0, or (x3-l){x= + 8a; + 10)=0; thus x2-l = 0, and xH8x + 10 = 0. 12. Here x«-7x2 + 12 + 2x(x2-4) = 0; that is, (x2_4)(x2-3) + 2x(x2-4)=0, or (x2-4)(x= + 2x-3) = 0; thus x2-4=0, and x'i + 2x-3 = 0. 1 9 13. Here s*=3x2 + 6x + 2, and therefore x* + x2+ =4x2 + 6x + -; 4 4 that is, ('''' + iy= ^ (2^ + 1)°- Thus x2-2x-l=0, and x2 + 2x + 2 = 0. 14. Here x« - 2x3 = i^^^ - lOx - 3, and therefore by adding - Sx'' + 4a; + 4 to each side, we have (x^ - x - 2)''=Sx^ - 6x + 1 ; thus x2-x-2=±(8x-l); thatis, x2-4x-,l = 0, and x= + 2x-3=0. 15. This is a reciprocal equation and may be put in the form 4(x + iy_20(x + l)+25=0; 15 11 whence x + -=z-. Thus the roota are 2, 2, -, -. XXXV.] THEORY OF EQUATIONS. 271 16. By inspection x = l is a root, and on reinoving the corresponding factor x-1, we have x* - 5x^ - 22x'' - 5x + 1 = ; that is, (a; + -) - 5 ( x + - ) -24=0; whence x + - = S or -3. \ xj \ xj X 17. The first derived equation is 4a;S + 27a;' + 24x - 80 = ; hence by Art. 559, this equation and the given equation must have a common root. Now the highest common factor of a;^ + 9a;3 + 12a;2_80j;-192 and 4x3 + 27x2 + 24a; - 80 is easily found to be x^ + Sx + lG or (x + 4)''. Thus x^ + 9x3 + 12x--80x-192 contains the factor x + i repeated three times ; the remaining factor is x - 3 ; hence the roots are - 4, - 4, - 4, 3. 18. If x*={x^ + ax + b)', we have 2ax^+{a' + 2b)x^ + 2abx + b^. By sup- position this reduces to the form x^ + gx + r = ; hence a' + 2b = 0, q = b, r=^. From these equations, \ve have ^ = -77 , and r= 3-; 2i 8 thus r^=%: = -%-, or q^ + Sr'^=0. b4 8 Suppose that 8x'-36x + 27=0 can be thrown into the form x*=(x2 + ax-6)2; then we have a^ - 26 = 0, and — = — -r = =r^ ; O DO Jil u 7 36 9 , 462 , hence 6=— = g, and a = — =3; these values satisfy the equation a^-2b = 0. Thus x*=(x'' + 3x-^ , thatis, x2=±/'x2 + 3x-g') ; 9 9 hence 3x-5 = 0, or 2x2 + 3x-^=0. 19. The required condition may be obtained by ehminating x between the two equations. [See Art. 528.] We have px2 + 25X + r = 0, and x^ + 2px+q = 0, whence by cross multipli- cation, x2 : X : l = 2(g2_j))') : r-pq :2(p^-q); hence 4 {p^ - q) (3^ -pr) = (pq - rf. According to the second supposition, the first expression is divisible by the second without remainder. Now (x'> + 3px^+?,qx+r) = {x+p)[x^ + 2px + q) + (2q-ip^)x + r-pq; hence 2(q-p^x+(r-pq) = Q for aZZ values of x; and therefore p'-q=0, pq-r=Q. T'huspr=p'^q = q^. 272 ' THEORY OF EQUATIONS. [CHAP. 20. By the conditions of the question, ax^+Sbx^ + 3ca; + d=0, and its first derived equation ax'' + 2bx + c = must have a common root; hence ix' + 2cx + d=0, and aa:^ + 2bx + c=0 must have a common root. Elimi- nating x^, we have 2(ac-b^x + a4,-bc = 0} whence x=-r; r^ ■ ^ ' '2{ac-b^j 21. We have x*+px^ + qx^ + rx + s=x^+px^ + qx^ + rx+ -^ =e-f"i)"-(i|-')-- Henoe (x^ + ~x+-\ = (?^- + -~ -q\ x^=a'x^, say; thus x^ + £x + -=±ax. 2 p 22. The equation whose roots are ±;^6-2 is a;^ + 4a;-2 = 0; henoe the other roots are given by a:*-4a;' + 8x-4=0. Thus x^~ix^ + ix^=ix^-8x + i, or a;'!-2a;=±(2x-2); that is, a;''-4x + 2 = 0, and x2=2. 23. Here2/=/3 + 7 + S+— =(a+/3 + 7 + S)-o + ^^; that is, y = 0-a + -=a[--lj; thus if X has the value a, then y has the value a{ — 1 ] ; and we have to substitute y = x( — Ij, orx(l-s)=sy in the original equation. Now x*{l-sY+qx^{l-s)* + rx(l-s)* + s{l-sY = 0; .: sY + is(l-s]''y^ + r(l-s)^y+{l-sY = 0. 24. We have a + p + y + S=p, aP + a.y + ad + Py + pS + yS=q, a/37 + "■P^ + "7^ + (37S = r, aPyd=s. (1) Suppose that o + /3=7 + S. In this case j5=2(a+/3); 3 = a^ + (a + ^)(7 + 5)+75=a/3+{a+/3)2+75; r=aj8(7 + 5) + 7S(a+;8) = a;3(a-l-j3) + 75{a + ^) = (o + /3)(a/3 + 7S). Thus 4g = 4 (a(3 + 78) +p^, and 2r =p (a^ + yd) ; whence we obtain ^q = 8T+p\ XXXV.] THEORY OF EQUATIONS. 273 (2) Suppose that 0^=75, In this case s = a^/S''', r=o^(7 + 5)+75(o + /3) = o^(a + p + 7 + 5)=i)o|3; hence r'=p''a''^^=ph. 25. Denote the roots by a and - ; then we have a' - 209a + 56 = 0, and a 56a^ - 209a* +1 = 0. Eliminating a^, we have 209a< - 56 X 209a + (56)= - 1 = ; but 56= - 1 = 57 . 55 = 19 . 11 . 3 . 5 = 209 X 15 ; hence a«- 56a + 15 = 0. Similarly by eliminating the constant from the two above equations and dividing by a, we have 15a* - 5&a^ + 1 = 0. From these last two equations, we find a3-15a + 4 = 0, and 4a3-15a2 + l=0; finally eliminating a^, we have a= - 4a + 1 = ; whence a = 2 ± ^3. 26. Denote the product of the two roots by y; then these two roots are given by the quadratic equation x^-5x + y = 0; hence i'-409a; + 285 must be divisible by a" - 5x +^. It will be found that the quotient is a;3 + 5x2 + (25-2/) a; + (125 -IO1/), and the remainder (2/2-752/ + 216)x + 5(2i/2-25?/+57), or (!/-3)(y-72)x + 5(2/-3)(22/-19). Thus the remainder vanishes when y = 3, and therefore the two roots are given by x= - 5x + 3 = 0. 27. If i=s/^ then {l + a'')(l + l}'){l + c'')... = (l+ia)(l + i6)(l + ic)... x{l-ia)(l-ib){l-ic)... = {1 -ipi + i^p^-i%+ ...) X {l + ipi + i'i)2 + v%+ ...) = {{^-Pi+Pi--)-i{Pi-Ps+P5--)} x{{l-p.2+Pt-...) + i(Pi-P3+P5--)} ^(l-P2+P4- -f+iPi-Pi+Pe- -V- 28. Tte given equation may be written (x2-4x + 8)2=x2-4x + 4 = (x-2)2; ' hence x''-4x + 3=J=(x-2); thatis, a;2_5^ + 5 = 0, or x2-3x + l = 0. If we put x=i-y, the above equations become y^-3y + l = 0, and y^-5y + 5 = Q respectively, and we merely reproduce the original equation. H. A. K. 18 274 MISCELLANEOUS EXAMPLES. [PAGE MISCELLANEOUS EXAMPLES. Pages 490—524. 1. If a is the first term and d the common difference, we have 2sj=n{2a + (rs-l)d}, 2s^=2n{2a+(1n-l)d}, 2s3=3n{2a + (3n-l)(e} ; hence ^ 4- -r-^ = 2 . =-^^ ; that is, 3s, +s,=3s„. n 3ft 2ft' '13 2 2. We have — ij-^ = — 7-^ = 94! tl^^* is, 3a; = 4j/, and a;!/ = 24 (a; - y). Hence 3a;^=24(4x-3a;) ; therefore (excluding zero solutions) x = 8, ^ = 6. 3. If r be the radix, 5r + 2 = 2 (2r + 5) ; whence r = 8. 4. (1) By rearranging, we have {x + 2){x- 4) {x + 3) (a; - 5) = 44 ; that is, {y - 8) (j/ - 15) = 44 ; where y = x^- 2x. We easily obtain ?/ = 4 or 10 ; hence a;^-2a:-4 = 0, or a;'' - 2a; - 19 = 0. Thus the solutions are l±^a, 1±2V5. (2) We have xy + xz= -2, —2xy + yz= —21, 2xz-yz=5. Solving these as equations in xy, xz, yz we obtain xy = '6, xz=-5, 2/2= -15; whence x)/0= ±15. 5. We have 2a + (p-l)d = 0. The sum of the next 3 terms = sum of (p + g) terms -sum oip terms =P±l{2a + (p + q-l)d}-Q. Thus the sum is {p + q)\a~ (^XJIf 1 = _ (y + g)ga p-X • 6. (1) One solution is obviously a; = 1. On reduction the equation becomes {a + b){ab + {a^ -b^)x — abx'} = a'x — a6^ + a'bx'' — Wx. The product of the roots = — ^ r-^ r- = - - — , : ^ a'b + ab(a + b) 2a + b' which is therefore the value of the second root. (2) If c = a + b; then c^={a + bf = a' + b^ + 3ab{a + b) = a^ + b^ + 3abc; that is, Sabc = c^-a^-b^. Hence the given equation is equivalent to 3^12a;(2a;-3){x-l)=12(a;-l)-x-(2a;-3} = 9(a;-l), or 12a;(2a;-3)(x-l) = 27{a;-l)3; whence a; - 1 = 0, or 4a: (2x - 3) = 9 (a; - 1)^ 491] JIISCELLANEOUS EXAMPLES. 275 7. We have (1 + d) (1 + 33d) = (1 + 9d)= ; that is, 4ScP - 16d= ; thus d = ord=i. 8. Here o+/3= -jj, a/3=j; hence and a* + a.^p^ + ^={a'' + ap + P'')(a^-a^ + fP) = {p^-q){p^-3q). 9. If 2.r = a + a-i, then 4.r^-4=a3 + 2 + a-»-4=(a-a-i)=. Denoting the given expression by £, we have iE = {a + o-i) (ft + 6-1) + (a - a-i) (6 - 6 -') = 2 (a6 + 0-16-') . 10. Without altering the value of the whole expression, we may douhle each of the expressions under the radical signs. Kow 8 + 2^15=t^'5+^'3V-. and 12 + 2^'35 = (^'7+^'5)«; hence the required value= ,."^^,. .. — ^^^ — ^7^^, (v''+^'o)■'-(^'7-s'o)^ ^ 5^/5 + 3 ■^/5.(^'S)=' ^ 5 + 9 ^_7_ 3(^'T)-\'5 + o^'5 21 + 5 13" 11. Replacing a and p by the more usual forms u and w', we have o* + 13* + o-i/S-i = u* + 0)8 + u-» = w + o,-^ + 1 = 0. 12. This follows from the fact that r'+2rS+4r2+3r+2 = (rS+r + l)(r2 + r+2). 13. Let X and y denote the nnmber of yards that A and B run iu a , ,, 1760-11 1760 .„ second: then =o7. y X . . 1760 „, 1760-88 y ^ To eliminate i, multiply the second equation by 20, and the first by 19, and subtract; thus - (20x 1760- 19 X 1749) = 20 X 81-19 x 57j or, - (1760 + 209) = 81 +450; hence y = -5- , and therefore x=^ . Thus A takes 420 seconds, and B 480 seconds. 14. See Ex. 4, Art 137. Thus from the first three equations we have _« ^ y _ - _,. a^-Vc* b*-(?a^ c'^-a?}^ Sabstitnting for x, y, j in j- + y+s=0, we get a4+64+cJ=6SgS+(J!aS+a2js. 18—2 276 MISCELLANEOUS EXAMPLES. [PAGE 15. Wehave (a-b)x^ + {b-c)xy + (c-a)y' = 0, or {x-y){{a-b)x- {c-a)y) = 0. Taking x=y, we have x'=y^= r — . Taking {a-b)x-{c-a)y, we fina-^ = -^ = J:, where ak^{a'+b^ + c^-bc-ca-ab) = d. 16. Suppose that the waterman can row x miles per hour in still water, and that the stream flows y miles per hour; then he can row x + y miles per hour with the stream, and x — y miles against the stream. Thus 48 48 , , , x + y x-y — — + = 14, and -j^ = -„-^ ; x+y x-y 4 d whence x=7y, and y = l, x=l. 17. (1) Theexpression = (a + 6)(a + c)x(6 + c)(6 + a)x(c + a)(c + 6) = (b + cY(c + aY[a + bY. (2) The expression = i {2 - 2a; + 2 ,^(5 - ix) {Ix - 3) } =i {JT^ii+ J2^^)\ TO ,u T^ ffi • ^. 1 10 7 4 1 2 5 „. 35 18. (1) Thecoefficient = jg..^.g. 3. 3. 3.3.3^=3-. (2) Wehave (|x-iy=.xs(!_^^y. . ■, . „ „ . , . 1 . Hence the term required is the coefBcieut of -jg in the expansion of the last binomial, and is therefore equal to 19. (I) W.l,„.(2-^)-(>-J^) + (l+^j)=0, -1 2 8 X — 1 X — ^ X — o whence 1x^ - 21x + 12 = 0, and x = ^^^v /"^ . 14 491] MISCELLANEOUS EXAMPLES. 277 (2) From the given equations, we have x^-xy-y^ _ —ab —1 (x + y)l^ax + by) 2a6(a + i) 2(a + b)' that is, (Sa + 2b)x'- (a + b)xy - {2a + b)y^=0. Thus x-y = 0, or (3a + 26)a; + (2a + 6)2/ = 0. If x-y=0, then from x^-y^=xy-ab, we find x^=y'^=ab. ^^ 2^" _ (3^+26) = ''^' *^®'^ ^°™ ('K+2/)(a^ + i!'2/) = 2a6(a + ^), ■we have - (a + 6) (2a2 - 2a6 - 26^) fcs = 2a6 (a + 6) ; that is, k^ {V + ab-a^) = ab. 20. When the expression is a perfect square, 4ac(6-c)(a-6) = 62(c-a)2; and therefore arranging according to powers of 5, we have 62(c + a)i'-4ac(a + c)6 + 4aV=0; that is, b{c + a)~ 2ac = ; which proves the proposition. 21. Since {y + z-2xy-{y-zy={2y-2x)(2z-2x) = i{x-y){x- z), we have (x-y){x-z) + {y -z){y-x) + {z-x){z-y)=0. Put y-z = a, z-x=b, x-y = c; then 6e + ca + ffl6=0, whUe a + 6 + c = 0. .-. (a + b + c)^-2{bc + ea + ab)=0; that is, a^ + ft^ + cS^O; thus 0=0, 6 = 0, c=0. 22. 3e5826i ( letS 1 2e ) 2e5 281 3tt ) 3482 _3304_ 3e85 ) 17(61 17t61 Let r denote the radix of the scale ; then , .. 1 A 7\ /, IN r + 7 thatis, _=(^_ + _j^(^l__J = -_; or r'^ - 5r - 36 = ; whence r = 9. 278 MISCELLANEOUS EXAMPLES. [PAGE 23. We know that 2(ab + ac + ad+... + hc + bd+...) = (a + b + c + d+...)^-{a^ + V + e^ + d^ + ...). From tliis the required result at once follows, since (l + 2 + 3 + ... + re)2=13 + 23 + 3' + ...+?i'. 24. Denote his weekly wages by x pence, and the price of a loaf by y pence ; then we have -- J = 6.andj^^-3| = lj; whence s=180, y = 6. 25. Denote the numbers hj a-3d, a-d, a + d, a + 3d; thus 4a =48, or a =12. Hence (12 - 3d) (12 + 3(Z) : (12 - d) (12 + (i) = 27 : 35 ; or 35 (16 - d2) = 3 (144 - d?) ; that is d^ = 4. 26. (1) By inspection, one root is unity; also the product of the roots . cla-b) .. ,, , ,. cla-b) IS ,, { ; thus the second root is —7; ( . a(b-c) a(b-c) (2) By an easy reduction we see that x H , =x-i , ; ^ ' x-a-b x-c-d that is ab(x-c — d) = cd (x-a-b). 27. (1) By transposing and squaring, we have a-x + b-x + i ij(a-x)(b-x)=c-x. Kepeating the process, we obtain (a + b-c-xY=4,[a-x)(b-x); that is, a'+h^ + c^-2ab-2ac-2be + 2{a + b + c)x-ix^=0, or {a + b + eY + 2(a+h + c)x-3x'^=i(bc + ca + ab). (2) Since x^+y^ + z^ = Zxyz when a; + 2/ + «=0, we have in the present case a + 6 + c = 3 Jabc; therefore (a + 6 + c)'=27a6c. 28. Suppose that the length of the journey is x miles, and the velocity of the train y miles per hour ; then LT. J. • 5(x-y) X , thatis, _J *.'__ = ! or a;=4«. 3y y ' " 492] MISCELLANEOUS EXAMPLES. 279 Again, the train takes IJ hours more in travelling 50 miles at the reduced speed than it does in travelling 50 miles at the original speed ; thus r— = 1J; whence !/= -Q-. Therefore x= -^. 5-' 29. From the first two equations by cross multiplication, we have 1 = ^ = 1=4 say; hence A;» (27 + 64 + 125) = 216 ; thatis, P=l. o i o 30. Suppose the two mathematical papers A and B were fastened together and considered as one. We should thus obtain 2 j 5 permutations among the Jive papers, since the mathematical papers themselves admit of two arrangements, and these oases are all ineligible. Also the whole number of permutations without restriction is 16; therefore the required number of arrangements is 16-2 15, or 480. 31. Let I, y, z denote the number of half-crowns, shillings and four- penny-pieces respectively; then x + 7j + z = G0. Also 30x-i-12i/-h42=1250; that is, 15x + Gy + 2z = 625. Ehminating z we have lSx + iy = 505 ; of which the general solution is x = l + 4t, and ^ = 123 -13«; hence « = 9t-64. 64 123 Thus t must be greater than — and less than ^p— ; that is, ( may have the values 8 and 9. Thus x = 33, y=19, z = 8; or x=37, i/ = 6, a = 17. 32. Subtracting the first expression from the second we have (6-a)x=-|-3x-l-2. Multiplying the first expression by 8, and the second by 6, and subtract- ing we have 2x {x^+{ia-Sb) x + 2\. Thus both {b-a)aP + 3x + 2 and x^ -I- (4o - 3i) X -I- 2 must divide each of the given expressions, multiplied if necessary by some positive integer. In these two quadratic expressions the term independent of x is the same ; hence the coefficients of x'' and x must be the same ; thus b-a = l, and 4(1-36=3; whence a = 6, 6 = 7. 33. Suppose that A, B, C together do the work in x hours; then A alone can do the work in x -I- 6 hours, B alone in x -)- 1 hours, and G alone in 2x hours. Hence working together they can do g -I =■ + — of the X -{- O X -J- X aX work in one hour ; but they also do - of the work in one hour ; 1 111 hence — -^ H r + ?;- = -; X + & x + 1 2x X that is, 2x(2xH-7) = (x-t-6)(x-l-l), or 3x2+7z-6=0. 2 Thus (3x-2)(x-^3) = 0; whence x=5. 280 MISCELLANEOUS EXAMPLES. [PAGE 34. Eliminating y, we have b^cx' + d(l-ax)^ = b'', or (b^c + a?d) x^ - ladz + d - 6= = 0. By hypothesis, this equation must have equal roots ; henco {;bH + a?S){d-V) = a?d?; that is, 62 (jsc + aSd) = y^U, or V^c + d'd = cd. Also the sum of the roots = ,s =-==23;; o^c + a'd ., , ad ad a -. , h therefore ''=b^^^^d = ^ = 0' »? symmetry 2, = - . 35. Here (1 - 2x + 2x^)~i = 1 + ^ (2a; - 2x^) + ^.|(2.-2.T+^.S.|(2-2.=)3 + |.|.|.^(2.)H... = l + {x-x')+~{x''-2x^ + xi) + hx^-3x*) + ~ai^+... = 1 + ^+2 -2 --8-+- 36. Denote the roots by o and a?\ then a + a''= -f, and 0' = }. Hence -J)3=a6 + 3a' + 3a^+o3=g2 + g + 3o3 (o^ + a) = 52 + 3-3253. 37. Arranging the equation in the form a;(a;2-l)-5 (a;2 + a!+l)=0, we have (aj^+a+l) (a;2_a;-5) = 0. 38. Subtracting numerator from denominator, we have x^ - 4x + 3, that is [x - 1) (x - 3). Hence numerator and denominator must be divisible by x - 1, or by X - 3, and must therefore vanish when x = 1 , or when x = 3. If x=l, we have a = 8, and in this case x3-8x2 + 19x-1 2_ x'-7x + 12 _ x-4 x3-9x2 + 23x-15~x2-8x + 15 " x - 5 ' If x=3, we find also that a=8. 39. This equation is equivalent to a2 + j,2 + c2-6c-ca-a5 + 3x2 + 3/ + 322=0, or i{{&-c)2 + (c-a)2+(a-i)2}+3x2 + 3)/2 + 3s2 = 0; and therefore 6-c = 0, c-a = 0, o-5=0, a;=0, 2/ = 0, 5 = 0. 493.] MISCELLANEOUS EXAMPLES. 281 40. With the notation of Art. 187, 3 6 + 4r - 4 .". 2V|.i>r„ so long as — = >1, or 2>Sr. Therefore the first term is the greatest. 41. Denote the numbers by X and 2/ ; then (x + y) (n? + y'^) = 5500, and (» - ?/) (a;^ - j,2) = 352 ; , (x + y)(x^ + y^) 5600 ^, ^ . x' + y^ 125 ^^"^"^ -, ; , i = -^^ , that IS , "— = -5- ; (x - y) [x^ - y^) 352 (x-y)^ 8 whence llTic^ - 260xy + Illy" = 0, or (13a; - Qy) (9a; - 13y ) = 0. Thus — = ^ =*say; and therefore 352 = 4i x SSi^ ; whence ft = 1. 42. From the data, x^+y^ + z'=V{a^ + b^ + c^) - 2X (6^ + Sc^) + 62 + 9^2 Jl + b^ + Sff _ 2(62 + 3e°)(l + 62 + 3a ^ a^ + ft^ + c^ a'^ + b-' + c'^ +0 +yc ^(l + 6= + W4^^ + ,.^9,. a- + i^ + c' _ 1 - 6^ - 66^cg - 9e^ + g'' (Jb" + 9c^) + (6" + c') (&^ + 9c^) " a' + b^ + c^ _ l + lbV + 9c^a' + a''b^ ~ a' + b'^' + c^ ■ 43. (1) Adda;2+4toeaohside; thena;H4a;2 + 4=a;2 + 16a; + 64; whence ar' + 2= ±(!«; + 8); that is, a;2-a;-6 = 0, or a;2 + a; + 10 = 0. (2) From the given equations, we have x^-y^ + x-y = 0, or {x-y) {x + y + l) = 0. Thus x=y, or x + y + l = 0. Similarly a; = «, or a; + x + 1 = 0. li x=y = z, we have 2rc^-x-l = 0; whence x=l or- j^. It x=y and a; + + 1 = 0, we have '2x^+x = 0; whence a;=0 or- -. If a;+y + l = and x=z, we also obtain 2x^+x = 0. Ifa; + 3/ + l = anda; + z + l = 0, we obtain 23;*^ + 8a; + 1 ; whence i = - 1 or - - . 44. log(a; + 2)+Iog{a;-2i/ + 2)=log{(a; + 2)2-2j/(a; + s)} = log {(a;+s)''-4a;2}=log(a;-2)'' = 21og(a;-£). 282 MISCELLANEOUS EXAMPLES. [PAGE ,^ , 11 1.3 /1\2 1.3.5 /IV 45. l + 2-5 + 2-T4ViJ +27476(4)+- that is, l + ls=^-; whence 5=1(2^3-3). 4 o o „ , „ . sum of numerators 5{x + y + z) 46. Each fraGtion= ^-= ^ — - — = tttt " sum of denominators a + o + c , , ,. (dx + 2y) + 2 (3y + 2z) + 3 (3z + 2a) Again, each fraction =4tt nrr — sttt^t — » . . d /■., o~\ ° ' (3a-26) + 2 {36-2c)+3 (dc-2tt) _ 9x + 8y + 1 3z _ , S{x + y + z) _ 9x + 8y + lBz ~-3a + ib + oc' a + b + c ~5c + ib-3a' 47. The first place can be filled in 17 ways, and the last place also in 17 vvays, since the consonants may be repeated. The vowels can be placed in 5 X 4 ways ; hence the number of ways =17x17x20=5780. 48. Suppose that at first x persons voted for the motion, then 600 - x voted against the motion, and it was therefore lost by 600 - 2x votes. Suppose that y persons changed their minds, then in the second case x + y voted for the motion, and 600-x-y against it; thus the motion was caiiied by 2 {x + y)- 600 votes. Hence 2{x + y)- 600=2 (600 - 2a;), and ^^ = y i whence a: = 250, y=150. 49. The expression on the left =— =— log {l + x) — log (1 - x) = -{log(l + 3i)-log(l-x)}-|{log(l + a;)+log(l-!<;)} / a;' x^ x' \ I x^ x^ x' \ = (^ + ¥+5 + 7+-)+n^'"4 + 6+-) =^+^'G+l)+-'6+^)+^K5+7)+ 50. Let X denote the number of men in the side of the hollow square; then the number of men in the hollow square =3;^- (a; -6)^= 12a; -36. Hence (12a; - 36) + 25 = {^x + 22)^; from which we obtain x - 4 ;^a; - 45 = 0, andx = 81. 495.] MISCELLANEOUS EXAMPLES. 283 51. (1) Divide throughout by v'a^^^; thus r/^+2 r/^=3; whence r/^±^= 1 or 2, and ^^= 1 or 2'". V a-x a-x (2) We have {x - a)i (x - b)i -{x- c)i {x - d)i = {x-c-x-a,p {x-d-x~b)i = {{x-c-x-a) (x-d-x-b)}^. Square both sides ; then {x~a]{x-b) + {x-c){x-d)-2{{x-a){x-b){x-c){x-d)]i = {x-a) (x-b) + (x-c){x-d)-(x-a) (x-d)-{x-b){x-e); hence (a; -a) {x-d) + (x-b) {x-c)-2 {{x-a) (x-b) (x-c)(x-d)}^=(i; that is, (x - a)4 (x-d)i-(x~ 6)i (x - c)i = ; whence, by transposing and squaring, (x -a){x-d) = {x- b) (x - c). -? -2 52. We have Sji = {2f = Q^ = (l - 1)' ; expanding by the Binomial Theorem we obtain the series on the right. 53. Put u=j6(5a; + 6)andi;=^o(tJa!-ll); then M-i; = l, and «'-w^=91. But u' - Suv {u-v)-v^=l; and therefore v,v = SO. From these equations we easily obtain u = 6 or -5, v = 5 or -6. Thus we have finally 6 (oa; + 6) = 216 or - 125; that is, a; =6 or - -^777. oi) 54. After the first operation the first vessel contains a-c gallons of wine, the second contains c gallons of wine. At the second operation x c gallons of wine are removed from the first vessel, and - x c gallons of wine are removed from the second vessel ; these quantities are equal if = =-, or c {a + b) = ab; that is, after the first operation equal quantities of wine are removed from the two vessels, and therefore the amount of wine in each will always remain the same after liny number of operations. 55. From the data, we have —7: — =zJab=. ; 2 ^ m + n 'henc6ma + nb = {m+n) ijab = 2 ijabxijab = 2ab; a,nim+n=2 Jab. From these equations we easily find m and n. 284 MISCELLANEOUS EXAMPLES. [PAGE 56. Let x + y + z = c, a constant. By hypothesis (c-3y){c-3z)=yz; that is, c^ -dc(y + z)+ 9yz = myz ; hence (9 -m) yz=c (Sy + 3z-c) = c {2y + 2z -x); thus 2y + 2z-x varies as yz. 57. We have and (l+a)-3=l-3a! + |^^x^-^a:3+... + (-!)'• ^'""^V('' + V +.... The given series is twice the coefficient of x^ in the product of the two series on the right; thus 4S = the coefficient of a;'' in (l + a:)"~^=""*C,; that is, S = 2x"-3Cr. 58. (1) We have identioaUy, (2x-l) - (3x -2) = 1 - x = (ix - 3) - (5x- i); dividing each side of this equation by the corresponding side of the given equation, we have j2x^-j3x-2 = Jix-3-j5x-i. By addition, ,j2x -l=;^4x-3; whence we obtain x=l. (2) Put a;2 _ 16 = yi, so that x^=y* + W; then 4 (2/3 + 8) =2/1 + 16 + 16?/, or 2/4-41/3 + 16^-16=0. Thus 2/' -16 = 42/ (2/^ -4); whence 2/^-4 = 0, and 3/2-42/ + 4=0; so that the values of 3/ are 2 and -2; and therefore a;^=32, and x= ±4^2. 59. Clearing of fractions we have {{y-z) + x{y'--z^)+x^yz{y-z)] + ... + ... = 0; that is, x{y^-z^+y{z^-x') + z{x^-y^) = 0; hence {y - z) {z - x) {x - y)=0, and two of the quantities x,y, z must be equal. 60. Denote the number of males and females by m and / respectively ; then m +f=p. Again Tfin + ■<()?, = T^' *^^* '^ bm + cf=ap. From these equations we have (b-c)m=(a-c) p, and (6 - c)f= (6 - a) p. Sag g.' 61. Ifi' = (^fj ,thena;»=U»)»^=^^j ; whence the required resultat once follows. 496.] MISCELLANEOUS EXAMPLES. 285 62. (l-x + a^-a;S)-i=, -1^ . - i±£=(i + ^) (l_x4)-i l-x + x'-x' 1-x* ' '^ = (1 + k) (1 + 35* + . . . + a;^"-H a;<" + a;*''+« + . . .). Thus the coeflSoient of a;*" is unity. 63. By simplifying each side separately, we have a{x-a) + b{x-b) _b {x-b) + a{x-a) ^ ab ~ (x- a] [x-b) ' hence the numerators being equal, the denominators must be equal ; thus a(x-a) + b(x-b) = 0, or [x-a){x-b)=ab. 64. If X is the common difference of the A. P., we have b = a + (n-l)x. Similarly if y is the common difference of the reciprocal A. P., T = - + (n - 1) u ; whence y = -y-, t- ■ b a ^ "" •' ab {n-1) Hente the r"" term of the A. P. =a + - '-\ '- = — '- — =-^ ^ ; »-l n-1 and the (ra-r + l)* term of the reciprocal A. P. _1 (n-r)(a-b) _a{n-r) + b{r-l) a ab(n — l) ab {n — 1) Hence the product required _a{n-r) + b{r-l) ab{n-l) m - 1 o (b - r) + 6 (r - 1) =ab. 65. Applying the condition for equal roots, we have p''(l + qy={p'-2{q-l)}{p' + 2q(q-l)}; that is, p^{l + qf=p''+p''{2q^-4:q + 2)-iq{q-lf; or p*+p^[q'-ioq + l)-iq{q-l)^ = 0; thus {p^-iq){p^+{q-i)'^}=0; and as the last factor is positive, we must have j)^-4j=0. We have {a + b)^=9ab; thatis, a + b=3^ab, 01 ^{a + b) = ^ab; hence log j- (a + 6) I = log (Jab) = 5 log (a6) = 5 (log « + log *)• 286 MISCELLANEOUS EXAMPLES. [PAGE 67. Let d te the oommou difference of the reciprocal A. P. ; theu -= +(n + l)d; ■whence (Z=- c~ a ac (n + 1) Hence the first and last means of the reciprocal A. P. are 1 a-c ,1 n(a-c) - + — ; n ' ^^^ - +- a ac{n + l)' a ac{n + l)' Thus the difference between the first and last mean of the H. P. / , i> f 1 1 1 ac{n + l){a-c){n-l) , (a + nc c + imj n-ac + n (a' + c^] + ac ^ provided that n'ac + n {a' + c') + ac =n^ - 1; that is, if n^ (1 -ac)-n (a' + c") - (1 + oc) = 0. 68. We have (^±^ll!i±lMrLzl) ^ 1 = 57 : 16; *v. *• 5718 that is, (n + 2)(n + l)n(n-l) = -j^ = 57.7.6.5.4.3; hence the product of four consecutive integers = 19.7.6.5.4.3 .3 = 21 . 20. 19 . 18. Hence n + 2=21, andn=19. 69. Suppose that £100 stock was issued at £x, then the actual rate of interest would be — x 6i . If the loan had been issued at £{x - 3), the rate of interest would have I, 100 .^ been 1 x J . „ 650 650 1 Hence = - ; X - 6 X 3 that is, 9 X 650 = 3 (a; - 3) ; whence x = 78. 70. From the identities {a + bf-a'-b^^ = Sab{a + J>), and {a-b)'>-a« + b3= -3ah{a-I)), we have {x^+x + lf-{x^ + l)^-x^=3x{x'' + l){x^ + x + l); {x^-x + lf-(x^+lf + x^=3x{x' + l){x^-x + l); {x*+x^ + l)^-{x*+l)3-x«=3x^(x^ + l){x* + x' + l); hence x'{x'' + lf(x''+x + l){x''-x + l)=x^{x* + l){x^ + x'+l); but x* + x^ + l = {x''+x + l)(x^-x + l); thus x = 0, x^+x + l = 0, x''-x + l = 0, and (x^+lY=x^ + l; whence the solution is easily obtained. 496.] MISCELLANEOUS EXAMPLES. 287 71. From the second equation, we have y{x + l)=-(lx + vi); hence by substituting in the first equation, {lx + m)--a(x + l){lx + m) + b{x + l)'=0, or x2(P -al + b) + x (2lm - aP-am + 2bl) + {m''-alm+hP) = 0. This equation is equivalent to iP+ ax + b — 0, l--al+b 2lm-al'-am+2bl vi'-alm + bP •^ i— = a = b • From these equations we have b{P- al + b)=m''-alm + bP, tliat is, al{b~tn)- {b^ - m^) = 0, or (6 - m) (al-b- m) = 0. Therefore either b = m, or b + vi=al. If we put b=m, by equating the first two fractions we obtain a{P-al + m) = ilm-al--am, or a-l-2a{P+m)+ilm=0; that is, {a-2l){al-2m) = 0, or a— 21, or al=2m. Thus either b = m and n =21, or b=m and al = 2m; and these last two conditions are equivalent to the single condition 5 + m = aJ which was obtained before. 72. (1) On reduction we have 3. 6=^-10. 6^+ 3=0; whence 6=== 3 or ^ ; _,_log3 ^47712 ^ „, . , X (2) On reduction, we have 10 . 5== - 29 . 5= + 10 = ; k1 5 2 whence 9 ""^ ^' X log5-log2 1-21ok2 thus ^ = ± 5 — 5 — =±T — i — .,-; 2 log 5 l-log2 79588 , , „„ , whence x= ±50097= * 1'139 nearly. 73. We have a;+j/ = 9 and 2:^ + j/' = 2417; hence ixhj + Gx^ + 4x!/»= 9< - 2417 = 4144 ; or xy{2x^ + 3xy. + 2y^ = 2072; but 2x' + 3xy + 2y^= 2{x + yf -xy = 162 - xy ; hence xy{l&2-xy) = 2072, or {xy - 14) {xy - 14S) = 0. The only admissible solution is obtained from x^ = 14 and x + y = 9, which givex = 7, y = 2. 288 MISCELLANEOUS EXAMPLES. [PAGe' 74. Suppose that n is the number of hours; then A has walked ll+4n miles, whUe B has walked S J9 + (n - 1) jl or — ^-tt — -^ miles. Thus "("+ •' -ll + 4n; thatis, n= + 3n-88=0; whence ra=8, 75. The expression (^3 + 1)''" + {JS - 1)'™ is an integer, and is therefore greater by 1 than the greatest integer in {^3 + 1)™, since (^3 - !)*» < 1. Hence the integer in question = (v/3 + l)=™ + (^/3-l)2'» = (4 + 2^3)"' +{4 -2^3)" = 2™[(2+;^3)"'+(2-^3)'»] _2m+ir2»' + 2™-2.^i^?i::i-).3 + 1; and is therefore a multiple of 2™+i, 76. The sum of the series 1, 3, 5, 7,... to x terms is x'', hence in the n groups there are n^ terms. It will be observed that the last terms of the first, second, third,... groups are 1^ 2^, 3^,...; hence the last term of the (n-l)"" group is (m-1)^; thus the first term of the m* group is (n- 1)^ + 1, and the number of terms in this group is m^- (»- l)^=2?i- 1. Therefore the sum = i^HlzH {2 (re - 1)^ + 2 + (2n, - 2) (1) } = {2n-l) {{n-l)^ + n} = 2n?-37i? + Sn-l. 11 113 77. We have (l-xf=l-\x-~x^- ^ ,^ ^ x^-...; also (l-a;)-^=l + a; + a;2 + a;3 + ...; By multiplying together the two series on the right, we see that the co- efficient of x" in the product is 1 - S; hence 1 1-5= the coefficient of x" in (1 - xf x (1 - x)-i \ = the coefficient of a" in (1 - x)"^ _ 1. 3.5.7. ■■(2n-l) ~ 2" [re 497.] MISCELLANEOUS EXAMPLES. 289 78. We have l±2g_ (l + 2.)(H-.) ^ 1^3. + 2.^ ^ 1-x + x^ l + a;S 1 + afl * i\ ^ J = (l + 3x + 2x2){l-xS + a;8+... + (-l)™a3"'+...}. If n = 3m, tlie coefficient of !i!"= ( - 1)™= ( - 1) ». ri-l If rt=3m+l, the coefficient of a;'»=3(-l)'»=3(-l) s . n-i If K = 3m + 2, the coefficient of ii!"=2(-l)™=2(-l) ». 79. (1) Putting a; = afc, y = bk, z = ck, we have , " ° — nr = ft; (a + + c) A whence fc = 0, or ° + ^ + '' . aic Thus E = | = ^ = 0, or^±^". a c abc (2) Equating the first two fractions, we have x^{!/-z) + y^(z-x) + z^{x-y)=0; that is {y — z){z-x){x-y) = 0. Putting 2/ -2=0, or 2/ =2, we obtain - + l + ^=x + 2y=3; y a; thus x^-2xy + y'=0, and x + 2y = 3; whence x=y = l. 80. The three arithmetic means between a and 6 are Sa+b a + b g + 36 ~r~' ~2~' 4 * Similarly the three arithmetic means between - and r- are a a + 36 a + & 3a + S 4a6 ' 2a6 ' 4a6 Hence we have ^ — ao — ^ = ^4 ! , 82g353 _ ^° (a + 36) (a + 6) (3a + 6) *' Multiplying these equations together, we find that a'6^ = 27, or ab = 3. Also (3a + 6)(o + 6)(a + 3&) = 240; that is (a + 6) {Sa' + lOab + SV') = 240, or (a + i){3(a + 6)2 + 4a6} = 240. Thus (a + 6)' + 4(a + 6) -80=0; whence a + b=i. Alsoa5 = 3. H. A. K. 19 290 MISCELLANEOUS EXAMPLES. [PAGE 81. Putting x-a = « and y- 6= », we have av-bu=ciju^ + v^; that is, {av-hu)^=e^{w' + v^), or (c2-62)»2 + 2ai!m» + (c2-a2)t)2=0. For real roots we must have a%' > (c^ - a') {c^ - b^) ; that is > c2 (c^ - a^ - 62) ; henoe c2 < a" + 62. 82. If (a! + l)2>5a;-l, then x^-3x + 2 or (x-2)(a;-l) is positive, so that X cannot lie between 1 and 2. If {x + l)^<:7x~'d, then a;^- 5a; + 4 or (a;-4)(a;-l) is negative, so that x must lie between 1 and 4. Thus a; = 3. 83. Since the logarithms of all numbers between 10" and 10"+^ have characteristic p, we have P = 10'^i- 10P=10»'(10- 1) =9 X 10". Again since the logarithms of all fractions between ^ ^ and r^ have cha- racteristic - q, we see that <3 = 10« - 10«-i=9 x 109"^ p Hence — = 10!'-9+i, ^nd therefore log P - log Q =2) - g + 1. 84. The number of ways is equal to the coefficient of x-" in the expansion of (a;'' + a;*-|-x* + ...)^; that is, to the coefficient of x^ ia {l + x+x''+ ...)^. This last expression is equal to r^ or (1 - a;)-*. •nr ^1, v f 5.6.7.8.9 ,„„ Hence the number of ways = - — = 12b. 85. Denote the sums invested by £x and £{x- 3500). The elder daughter receives the accumulated simple interest on £x for 4 years, the rate of interest being £4 on every £88 ; hence she receives £xx4xl. Similarly the younger daughter receives £{x - 3500) x 7 x ^ ; ^^. 2a; a; -3500 , „„„„ thus YT = 5 — ; whence a;=7700. 86. In tte scale of 7 let the digits beginning from the left be x,y,z; then 49a; + 73/ + z = 8l2-|-9i/ + x; thatis 24x-2/-402 = 0; or i/ = 8(3a;-52). Now 2/ must be less than 7, and 3a: - Sz is an integer ; hence 3a; - 52 must be equal to zero, and therefore ^ = 0. Again - = 5 = & say ; and thus a; = 5A;, o o 2 = 3ft. But a; and 2 are both less than 7 ; hence ft = 1 ; that is, a; = 5 and y = 3. 498.] MISCELLANEOUS EXAMPLES. 291 87. The sum of m + n terms, and the sum of m+p terms are each double of the sum of m terms ; thus ^^ {2a + (m+n-l) d} =~^ {2a + {m+p-l}d} =m. {2a + (m- 1) d\ =s suppose. Therefore 2a + {m,+n-l)d= and 2a + (7«- l)d = - : m + n' ^ ' m' whence „d = s T^- - i^i =i^iz:?li-. \m + TO mj m(iii, + n) Similarly pd=i!!!l^. m (7ft +p) Hence - = "'("'-") ('" + P) . ^j. (m + »){7n-p) ^ (m+p){m-n) p m{m + u)(m-p)' lup mn 88. Putj/-2=u, z-x = v, x-y = w, so HixsA u+v + w = 0; then 1 i =0. vw wu uv „, 111111222 Thus _++_=+++_ + — +_. u- v' w^ w IT w^ vio WU uv 1 1 1 /I 1 1\2 or ++=_+_ + _. U' V^ W- \U V ID J l'» + 3'" + ... + (2n-l)"» n+3 + 5 + ... + (2n-l)l'» ^^ ^ . ^ —> i ^ y ; that IS, >»">. n [ n I 89. 90. Suppose that the three equations are equivalent to {x-/3)(a;-7)=0, (x-7){x-o) = 0, (a;-a)(x-|3) = 0; then P + y=Pi, y + a=P2, a + p=p^; Thus pi'-iq^={p-yr={p.,-Ps?; that is, iq^=Pi'-p,'-p,'' + 2p,p,. Hence Mgi + qi + ist^^iPsPa+PsPi+PiPi) -Pi^-Pi-Pi- 91. Let X = the common rate of A and B in miles per hour, and suppose that B starts y hours after A. Then when J is at i or at any previous instant B is xy miles behind A. 3 Now the rate of approach of B and the geese is x-- miles per hour; therefore we may say that at this rate xy miles are covered while the geese go 5 miles at miles per hour. Hence ^ = ^ (1). '- -J. 19—2 292 MISCELLANEOUS EXAMPLES. [PAGE Again when A meets the waggon, B is xy miles behind, and A and the waggon are 50 -2a; miles from L. When B meets the waggon, he is 2 31 + 5 a; miles from L. Therefore the waggon has travelled in the interval o ( 31 + - a; J - (50 - 2x) miles. And since the rate of approach of B and the 9 waggon is a: + 2" miles per hour we may say that xy miles are covered at this 8 9 rate while - a; - 19 miles are covered at -r miles per hour. ^x-19 Hence J^=i—— (2). =^ + 4 i By equating the values of xy from (1) and (2), we get a simple equation in x 25 which gives x = 9 ; whence 2/ = ■jr ^^^ xy = 25. 92. Since d= - (a + 6 + c), we have abc + bcd + cda + dab = abc- (bc + ca + ab){a + b + c)= - {b + c)(c + a)[a+b) = ^/(a + 6)(o + c)(6 + c)(i+a)(c + a)(c + 6). Now {a + b){a + c) = a(a+b + c) + bc = bc — ad; hence the required result follows at once. 93. For the A. P. the common difference is b-a; hence the (n + 2)"' term is a+(re + l)(6 -a)= -na+{n+l)b. For the G.P. the common ratio is -; hence the (n + 2)"' term is a " \a) ~ IF ■ For the reciprocal A.P., the (re+2)"' term is 1 — = — ; and therefore . ab ■ ' {« + l)a-nb' When the three means are in G. P . we have {■ -na + {n + l)li]ab 62»+2 {n + l)a- -nb a^" ' tis. -na + {n + l)b ftsn+l (n + l)a-nb a^n+i > (» + 1) { 062"+! - o*»+i b]=n (62»+2 _ a»i+2) . .] MISCELLANEOUS EXAMPLES. 293 94. We have -^ f^ =-l-(^ ^ ) (x — a){x-b) a — b\x — a x-oj 1 / 1 1 \ a^ "b"- Thus the coefficient of a:" is r ( ;; + r; 1 = „.„ , , , • a~b\ a" 6"/ aH^{a~b) Wehave (1±£T ^ {(l^M" . Expanding the numerator by the Binomial Theorem, and dividing each term of the expansion by (1 - x)^, we obtain (1 _ x)=»-3 + 2nx(l- a;)*'-^ + . . . + " '" ~ ^^ {1 - x) (2a;)''-2 Hence the coefficient of rr^" must come from the last two terms, and therefore is equal to n2'^i + "*""'" J '" "*" ' or 2"-i (n2 + 4n + 2). 95. Wehave 15a;2-34a;2/ + 15!/'=0; whence (5a; - 3j/) (3x - 5j/) = 0. On reduction, the first equation gives 2{x-y) + Jx- - i/^ = 2 (x - 1) ; that is, Jx"^- !/2=2 (?/-l) ; whence x^=5y^-8y + i:. Putting 5x=Sy, we have 9y^=25 {5y^-8y + i), or 292/2 - 50j/ + 25 = ; whence 29!/ = 25±10 v/^. Putting 3x=52/, wehave 25)/2= 9 (52/2 -8?/ + 4), or Si/'- 18!/ + 9 = ; whence « = 3 or - . 96. Let X denote the value of the continued fraction; then 1111 1 1 a;-l = 3+ 2+ 3+ 2+ ■■■ 3+ 2 + x-l' 1 1 x + 1 *^*"' '^-^=3TiT1^3iT4' or 3a;'' + a;-4=x + l, and 3x^-5 = 0. 97. The first part easily follows from Art. 69; but may be proved directly as follows: -If 4 -n^ . n=n^ |(^)= - ("-^)} = ^^ - ^,: 294 MISCELLANEOUS EXAMPLES. [PAGE This holds whether n is odd or even : but if n is odd, we also have which shews that there is a second way. FinaUy {n + l)'-n^=3n' + 3n+l=k say; but k=(^^ ~ (^^Y' and k = Sn{n + l) + l, and is therefore an odd integer, since m(n + l) is even; k+1 k~l hence both — jr— and —5- are integers. 98. Hera 2S=|+ i+ i|+ ^ + ... 3-1 5-1 7-1 9-1 ~[3"'"|5"'"[7"'"|9"'"'" _J._ J. J._ 2 }:_2: -|2-^+|_4-15+L6-|7+- 1 99. We have x=- , andy=-= ; hencexy + hx = a, ani xy + dy=c. By subtraction, bx--dy = a- c. 100. Assume for the scale of relation 1-px- qx'^. [See Art. 324.] Let S=l + 5a; + 7a;2 + 17a;3 + 31a^+...; then -pxS= -px-5px^-7px^ -llpx*- ...; — qx^S= —qx^-5qx^-Tqx'^—..., .: S (1-px- qx') = l + {5-p)x; the quantities p and q being given by 5^ + 2 = 7, and 7p + 5q = n; whence p = l, 3 = 2. Hence S = - — ^^r^, = ,— V - :; ; aii2. 500.] MISCELLANEOUS EXAMPLES. 295 (2) We have V{a-c)''= b^ (w> - iae + c^) = 262(a2 + c2)-4aV = 2 (62 (a2 + c2) + 2aV - 2aci {a + c)}, Binoe 2ac=5(a + c). Thus 62 (a _c)2=2{c2 (6 -a)2 + a2 (0-6)2}. 102. The given expression vanishes both when x = a, and a; = 6 ; hence a3_3aft2 + 2c3=0, and 2c3- 263=0. This latter equation gives b = c, since by hypothesis 6 and c are real ; and therefore a»-3a62 + 263=0, that is, (o- 6)2{a + 26) = 0. Hence a = b, or a=-26. 103. Denote the numbers by 2m -1, 2n + l, 2ra + 3. Now l + (2n-l)2 + (2m + l)2 + (2rt + 3j2 = l2(n2 + m + l); but re^ + „ jg even ; hence n^ + n + 1 is odd, and the sum is an odd multiple of 12. 104. We have ax' + 2bx + c = al x + - ] H : \ aj a if therefore a is positive, is the least value of the expression ; if o is negative, it is the greatest value. From the given equation, we have (a;2- 2/«)2+ (3/2 - zx^+iz"^ - xyY=(i; hence x^-yz = 0, y^-zx = 0, z''~xy = 0; and therefore x^ + y^ + z^-yz-zx-xy = 0; that is, {y-zf + {z-xf + {x-yj^=0; whence the required result at once follows. 105. By inspection the value of the expression =-^--;5 *^-^r — ; and the required result follows at once from the Binomial Theorem, since |_ 1 _1 £2 ]_3 ^3 1.3.5 a* 1.3.5.7 a;5 (1 + x) -l + ^x 2-4+2. 4'6 2.4.6- 8 +2.4.6. 8'i0~- 106. We have a + /3= -p, and a^ = q. Also a2"+i)''a'' + g = 0, and ;32"+p»;8"+e = 0; whence a2"-|82» + p''(a'*-^») = 0, or a" + ^'»+p"=0 ; thus o" + j3"+ (a + /3)''=0, since n is an even integer; a /9 and therefore a;" + l + (ii; + l)"=0, where « = - or -. ' So 296 MISCELLANEOXTS EXAMPLES. [PAGE 107. Denote the values of the continued fractions by x and y ; then x-a= ; r; whence a;^=a' + 6. 2a + (x - a) Similarly y^=c^ + d; thus x^-y''=a? + b -a'-d. 108. Let n be the number of persons; then the number of shillings that the last person receives is l + l + 2 + 3 + ... + (m-l), orl+-&^; therefore ■'■+"9 — ^ = 67; whence n-12. The number of shillings distributed =n+ - Sn(?i- 1) = re+g(7i + l)n(ra-l) = 298, since «= 12. 109. (1) The equation is obviously satisfied by x = a, y = 'b, z^c, and being of the first degree there is only one solution. Or thus, (b + c)x + ay + az = 2a(l) + c), }3X + (c + a)y + bz = 'ib(o + a), cx + ci/ + (a + 6)z = 2c(a + 6). Adding the first two equations together and subtracting the third from their sum, we have bx + ay = 2ab ; that is, - + |=2. a b Similarly we may obtain the equations f- + - = 2, and - + - = 2; DC a c 1 "' y z . whence - = f = - = l. a b c (2) Clearing of fractions, we have 3{x^ + y'')(l + xy)-4.0xy, and lOx?/ (1 + xH 2/=) = 33 (a;' + )/). From these two equations, The case xy = may be excluded as the equations are not then satisfied. Itxy = 3, then x^+y^ = 10. 500.] MISCELLANEOUS EXAMPLES. 297 110. Divide by a-b; tlien it mil be sufficient to shew that n-l on-i + a''-2 ft + a»-»&2 + . . . + a6»-» + i"-' >n{abj^ . This readily follows from the inequalities, 71-1 n-l an-1 + Ji^-i > 2 (ab) ^ ; a^-% + ab"- ^ 2 {ab) ' ; and so on. a''-362 + a26'i-3>2(a6) " ; [Compare Ex. 89.) 111. Performing the operation of finding the greatest common measure of 396 and 763, we have 1 ; 12 1 9 1 396 763 1 29 367 1 10 19 1 9 thus 763 1 1 1111 396~ '^'1+ 12+ 1+ 1+ 1+ 9' „, . , 1 2 25 27 52 79 The successive convergents are tit.ts, Ya> o?' 77" Hence 79 . 396-41 . 763 = 1, and therefore 948 . 396 - 492 . 763 = 12 = 396x - 7632/. X - 948 u - 492 Thus -" whence = t say; 763 396 !<:=948 + 763t; j^=492 + 396t. 112. Suppose that A, B, C working alone would do the work in x, y, s days respectively. Then, since B'b and C'b joint daily work is m times ^'s . ■, , 1. roll daily work, we have — = - + -. m + 1 111 n + 1 p + 1 . . ■ , = - H 1- ^= = ■ Similarly; X X y z y z ■^ Therefore which proves the first part of the question. 1 1 /I 1 1\ Again ^ = --^- + - + -1: ^ m + 1 X \x y zj' 1 1 1 _^ " ro + 1 n+ 1 p + X~ And m + 1 = 1- 1 m+l' TO n p. /I 1 1\„ ■'m + l n + 1 p + 'i. \m+l n + 1 p + lj~'" 298 MISCELLANEOUS EXAMPLES. [PAGE 113. Let £(7 denote the constant expenses, B the number of boarders, and £P the profits on each boarder ; then since each boarder pays £65 we have, B (65 -P) = C+m£, where m is some constant. If £ = 50, then P = 9 ; hence 2800 = C + 50m ; If JS = 60, then P=10f ; hence 3260 = + 60m; whence m=46, C=500. Putting B = 80, we have 80 (65 - P) = 500 + (80 x 46) ; whence P = 12|. 114. We have y = ^-—^; hence 1" 2/^=( fljl^a ) • Taking logarithms, we have -log(l-2/'') = 2{log(l + ;r!')-log(l-:r«)}; thatis, y2 + y^ + y^+ ..=4J^2 + J + ^^+...j. 115. We have x {a^-x^)=y{a^~i/); thatis, {x-y){3i^ + xy + y''-a^=0. (1) Taking x-y = 0,-we have a; = y= ±c; hence from the equa- tion bx=a'- y', or bV = (a' -y^)^, we have 6'^c- = (a^ — c^)^ (2) Taking x^ + xy + y''=a^, and combining it with xy = c', we have a:^ + / = a--c2. Again we have -; — —. = -; ora; + u = 6, since x-y is not zero. x' -y' .: x' + y^ = V^- Ixy = h^- 2cK By equating the two values of a;^ + j/^ we obtain o'' + c'' - B- = 0. 116. The first result follows at once by putting x=-l in the first of the given relations. Multiplying together the two given expansions, we see that series (2) is the coefficient of a^ in the product and is therefore equal to the coefficient of x^'' in {(x^ + x + 1) (x - 1) y^'', that is in (x^ - Ij^"" ; this is equal to the coefficient |3r of ^' in (2/ - 1)» and therefore to ( - 1)'' T^pT^- 117. (1) From the second equation we have {x-a)(y-b) = Q; hence x = a or y = b. Substitute x = a in the first equation ; then (a-yY + 2ab = a^ + by, or y'-2ay-by + 2ab = 0; whence y = b, or 2a. Similarly if y = b, then x = a, or 26. 501.] MISCELLANEOUS EXAMPLES. 299 (2) Prom the Beoond and third equations, we have 2y^-zx = 13-iy, or zx = 2y^ + iy - 13 ; multiply this by 2 and add to the first equation; thus (x + zf-y'=iy^ + 8y-20; and therefore suhstituting from the third equation, we obtain (2 + 2/)' -1/2 = 4^3 + 8?/ -20, or y'' + y-Q=0; whence y=2 or —3. Substituting these values of y, we find x^ + z^=10, and x + z = i; or x^ + z'' = 15, and x + z= -1. 118. By taking the n letters in pairs we can form — ^- — inequalities of the form 2 iJajO.^ <: oti + a^. By adding these together we obtain the required result, since in the sum each of the n letters will occur on the right-hand side n - 1 times. Thus 2 Ja^+2 Jaiag + ...<:{n-l) {ai + a2 + ...+a„). Divide both sides by n (m - 1) ; then fja^z + V^i^s + ■ • • to '^-^ — - terms gi + a2+... + g„ n{n-l) n 2 which proves the second part of the question. 119. We have b^x* + ah/*=a^1i' (x^+y^; that is, b^x' (x" - a') = a^ (6' - V'')- Buta;2-o2=62_2,S; hence b^x^ = a?y^, and ' {b^x^-a'r)' = 0, or b*x*+aY=^a^bVyK Now b*x^ + aY = (6"^;'' + aY) (^'^ + V') = 6%8 + aV + 2a-Wa^^ = (6V-|-ay)2. 2r + 1 1 1 120. (1) Here -_^^^ = _ - ^--^; hence the sum = 1 - (n + lf (2) The series is the sum of the two series, a (x"-i + x"-" + . . . + ac + 1) ; and 6 (a;''-i + 4x''-2 + 9a:'^3-l-. .. + »-). 300 MISCELLANEOUS EXAMPLES. [PAGE The second series is a recurring series whose scale of relation is ( 1 — ] . 3 3 1 If we multiply the expression in brackets by 1 1'~2""3'^^ shall find that the first terms of the product are x'^^+a;""^, and that the other terms are zero with the exception of some at the end. Also the coefficient of i= - Sn^ + 3 (n - If - (n - 2)== - (m+ 1)= ; the coefficient of -^=3n^-{n-lf=27v' + 2n-l; the coefficient of -= = - n\ .: g=a^^+ ^ {a"+8 + «"+'- (m + l)ga:^ + (2ro^ + 2n-l) a- n°}. Uj ^ ^ \X ' — -1-} 121. ^"t g^'^+tf+e ^^' then2i/i=i + (3t,-l)x + 62/-2 = 0. If X is real, (^^J-lf>6y[&y~2)■, that is, 1 + lOy - 392/^ or (1 + 13y) (1 - iy) must be positive ; hence y cannot be greater than 5 . 122. (1) On reduction the given equation becomes 3x4 + Ux' + 21x2 + 14x + 3 = ; which is a reciprocal equation. Putting x + -=z, X we have 32^ + 14z + 15 = 0, or (3« + 5) (2 + 3) = 0. Thus 3x2 + 5x + 3 = 0, or x2 + ax + l = 0. (2) We have 3xy=-2z, x«=-6y, 2j/z=-3x; hence by multi- plication, x'4/22''= —6xi/«; and therefore xi/2=0 or xyz=-^. The given equations are clearly satisfied when x = 0, !/=0, 2=0. If xyz= -6, we have 3x2= -2x2/2 = 12; also (iy^= -xyz = 6; and 2z2= - Sxyz = 18. 123. Suppose that Oj, a^, a^, a^ axe the coefficients of x', x'+i, x'^^, x"^^ in the expansion of (1 + x)"; then Ol + Oj 502.] MISCELLANEOUS EXAMPLES. 301 Similarly 5— = ; and — ^— = r ; a^+ai 71 + 1 O2 + O3 n+1 whence the required result follows at once. aa + Ta^-a-S _ ^g + B , Cx + D (i2 + a + l)(a;2-3x-l)~a2 + a. + iT^a;a-3i-l* then a!3 + 7a!''-a!-8 = (4a+B)(af'-3a!-l) + (Ca+D)(a!a + a: + l) (1). Put«'>=3a: + l; thenK' + 7a2-a-8=a(3a + l) + 7!c2-K-8 = 10a?-8=30a; + 2; and [Cx + D)(x'' + x + l) = (Gx+D)(ix + 2) = 4C (3x + 1) + 4Dx + 2Ca; + 2D = (14C + 4D) X + (4(7+ 2D). Therefore 30a; + 2 = (14C? + 4D) x + (4C + 2D) ; that is, 7C + 2D = 15, and2C7 + D = l; whence ^ 13 ^ 23 Also by equating coefficients in (1), A + C=l, and -B+D= -8; whence 4= --5-, and £ = 5. o o x3 + 7x2-x-8 1 13X-23 1 lOx-1 Thus (xi'+x + l)(x2-3x-l) 3"x2-3x-l 3'x2 + x + l' (2) ^-4-i^=i(^»-«)(^-ir TT ,.1, ffi- X , r IJ^)- 8(r + l)1 6r-8()- + l) r+4 Hence the coefficient of x'^= j -^^F^i \r \ ~ 21-+2 2''+i ■ 125. If tte scale of relation is 1 -px - qx^, we have ^=-2^ + 19' '=2p-22: 5=ijZ + 2g; 7=5^ + 3?. From the first two equations, 9p = 5l+i, and 92 = 2^ + 16; hence 45 = J(5Z + 4) + 2(2J + 16), or 5P + 8i-13 = 0; whence J=l or - — . The value 1=1 is the only one which satisfies the fourth equation T=5p + ql; thus i=l, ^ = 1, 2 = 2, and the scale of relation is 1 - x - 2x^ 5_7 Hence the generating function =5^^^ = 5 (f|^ + J^) ; and the general term= j {2"-^ + 4 ( - l)»-i} x""! = {2»-3 + ( - 1)"-^} x""'. 802 MISCELLANEOUS EXAMPLES. [PAGE 126. From the first two equations, we have or 2a-3(i/ + z) = 2x-y -a; that is x+y + z = a. Henoe 2a-3 {z + x)=2y-z-x; .: 2a{z-x)-3 (z^-x-) = 2y [z-x)- (z^-x^). But 2az-3z^ = {x-yf; hence 2ax -3x^={x-yf- 2y {z-x)+z^- x'=(y- zf. 127. (1) We have a;2 + a;j/-2:i; + 6 = 0, and xy + 1/2- 2y- 9 = 0; hence by addition, (a;+y)'-2 (a; + j/) - 3=0; from which we find x + y=3 or -1. By subtraction, we have (as - 1/) (a; + 1/) - 2 (x - 2/) + 15 = 0. If x + y = 3, we have 3 {x-y)-2 {x-y) + 15 = 0, or x-y= -15; whence a;=-6, y=9. If x + y=-l, we have - {x-y)-2 (x-y) + 15 = 0, or x-y = 5; whence x=2, y=-3. (2) Taking logarithms we have log a (log a + log a:) = log 6 (log 6 + log I/) , and log 6 log a; = log a log J/. For shortness put log a=^, &c.; then we have AX-BY=B''-A'^, and BX=AY; whence X= -A, and Y= -B; or loga;= -log a, and logi/= -log5; thus a;=-, y=T- , , x2(a;2 + a2)-(a:4 + a4) 128. (1) xjx^^a'-j^^a^=^^^j=^^^j=^^ _ a'x^-a* _ X Jx^ + a^+ Jx* + a^ and when a;= oo this becomes — = , = -„- . x^+x' 2 .q. Ja + 2x- ,j3x _ (a+ 2a-) - 3x j3a + x + 2 ^Jx J3a + x-2jx~ Ja + 2x + j3x' (3a + ar)-4j: 1 j3a + x + 2Jx 1 4 2J3 , = 77 • . ^= = r • Pi — ,„ = — /— ) when x = a. a J:;:^x + J3x 3 2V3 9 603.] MISCELLANEOUS EXAMPLES. 803 129. Let X and y denote the two numbers ; then xy = 192. Let g denote their greatest eommon measure and I their least common multiple; then B|,=€±-^ M J.^/. Nowsr?=xy = 192. [See Elementary Algebra, Art. 163.] Thus ^ (p + J)2=169xl6; so that g + l = 52. Also gl =192; hence 3 =4 and l = i8; that is, the greatest common measure is 4, and the least common multiple is 48. The numbers may therefore be denoted by 4p and 4j where p and q have no common factor. The least common multiple is ipq; hence pq = 12, and therefore p = 3, q = i; orp = l, q = 12. 130. (1) If a-ft = c, then c3=a'-6'-3a6(a-b) = a3-6S-3a6c; that is, 3a6c = a^-6' — c'. Thus 34/2. ;>yi3x + 37 ^13.; - 37 = (13a; + 37) - (13x - 37) - 2 = 72. By cubing each side, we have 169x2-1369 = 6912; that is, X- = 49, or X = ± 7. (2) Multiply the first equation by - a, the second by 6, and the third by c and add; then 2bc ^l-x-=b'' + c'-a^; that is, 46V(l-x=) = (62 + c2-a2)2, or 46 Vx= = 26 2c2 + 2c^a^ + 2a^b^ - a* - 6< - c*. 131. Wehave 2J=(l + l)i 3 2'2 2"2"2 2'2"2'2 2"2'2'2"2 = 1 + ^ + -!:; iT-+ -: rs + •••• 2 ^ li li £ Therefore 2J2 = l + ^ + --3S=~-3S. 132. Let r be the radix of the scale and suppose that when the number ar' + br + c is multiplied by 2 the result is cr'^ + br + a. Then re- membering that a, 6, c must aU be less than r and that c must be greater than a, it easily follows that 2c=r + a, 26 + l = r + 6, 2a + l = c. Prom the first and third of these equations we see that {ar + c)x2=r{c-l)+r + a = cr + a. Again, r=2c-a=2 (2o + l)-a = 3a + l, and only one out of every three consecutive numbers can be of this form. 304 MISCELLANEOUS EXAMPLES. [PAGE {l-x){l + x){l-x) \ 1-x J \ 1-xJ = l-1>2x^{l~x)~^ + x*(l-xy. Hence the coefficient of a;'" = 2 + (r-3)=j--l. 134. Let X, y be the number of yards in the frontage and depth of the rectangle ; then 3a: + 2y — 96, and we have to find the maximum value of xy subject to this restriction. Now 96 X 96 = (3a; + 2!/)2 = "iixy + (3a; - ^yY, and therefore xy is a maximum when 3a;- 23^=0, and the value of xy is then 96 x 96h-24, that is 384. 135. The expression is of four dimensions, and obviously vanishes when a=0, 6=0, c = 0, d = 0, and therefore must be equal to kabcd. Putting a=b = c = d=l, we have /c = 4^-4 . 2^=192. 136. Assume x^ + an^ + bx^+cx + l—lx^+^x + l) , and x* + 2ax3 + 2bx^ + 2cx + l = {x' + ax+l)^; then by equating coefficients we must have b=j + 2, c = a, 26=a2 + 2, 2c = 2a. Thus -5- + 4=a2 + 2, that is, a2=4; hence a=±2, c=±2, 6 = 3. 137. (1) After multiplying up and transposing, we have ^x + y=-2 ilx-y; that is, (x + y)= -&(x-y], 01 9x=7y. Hence | = | = *. where 130i= = 65, or ji;^ = 1 . (2) We have identically {2x^ + 1) - (2a;2 - 1) = 2 ; hence by division JE^+T - J2x^ - 1 = ^3 - 2x^ ; .-. 4a;2-2^4a;4-l = 3-2a;2, or i(2x'~l) = JIi^iZl; :. J 2%^ -1 = 0, or 3 sj23^^ = JW+l. 504.] MISCELLANEOUS EXAMPLES. 305 138. Suppose the number of pounds received for the first lot is expressed by the digits x, y ; then the price of each sheep = — r^ pounds. The number of pounds received for the second lot is expressed by the digits y, X ; and the price of each sheep is — ^^^ pounds. o _,, 10a; + M lOw + x 1 Thus -^^ ^ — = ^; that is, 8x-19y = 5; whence x=3, y = l, since x and y are each less than ten. 139. (1) Thesum = 2n(l + 2 + 3 + 4 + . ..)-(!. 1 + 2. 3 + 3. 3 + 4. 7 + ...). Nowl.l + 2.3 + 3.5 + 4.7 + ... = Sn(2n-l) = 2Sn2-Sre _7i(n + l) (2« + l) n{n + l) ~ 3 2 • Hence S= ^" • " ^" + ^) _ nln + l){2n + l) _^ n{n + l) 2i o 2 n(7i + l) L 4rt + 2 ,] B(re + 1) (2)1 + 1) =-4-- r-^-+i[ =~ — I — ■ (2) The general term of the series is ^ , and we have to a find the value of S "'^"Z^)" . 4 Now n2(re+l)3=n(n + l){(« + 2) (n + 3) -4 (re + 2) + 2} ; hence 4S=-n (ji + l) (n + 2) (re + 3) (7i+4)-7i(n + l) (n + 2) (re + 3) U 2 + g)i(?i + l){ji + 2) = TTn(n + l)(™ + 2)(3n2 + 6n + l). lo (3) The general term of the series is - (2n - 1) In, or n (2k - 1). „ „ B(n + l)(2n + l) n(m + l) 1 , ,.,, ,, Hence S=— ^ (-^ -' ^—= — ^ = ^n(n + l)(4n-l). 140. Proceeding as in Art. 526, we have identically l + 2t/2+rj/3=(i_„2,)(i_py)(l_^y). Take logarithms and equate the coefficients of powers of y ; then D? + ^ + -f_ a3 + ^3 + y) _ a< + /3^ + 7^ _gg a° + ;3° + 7^ _ 2 "*' 3 "'■' 4 -2" 5 "^'■' from these equations the required result at once follows. H. A. K. 20 306 MISCELLANEOUS EXAMPLES. [PAGE 141. (1) Substituting for x from the first equation, we have 27 8 y Sy-S + V; 15 that is, 21i/»- 1082/+ 135=0, or 7y''-3Gy + 45=0; whence 2/=3ory. (2) From the first and third equations, we have x^ + y''' + z'-3xyz = 180; dividing this equation by the second, a? + y^ + z^ - yz - zx - xy =^12. Subtracting this equation from the square of the second, we obtain yz + zx + xy==71. Thus x + y + z = lS, yz + zx + xy = ll, xyz = 105; hence x, y, z are the roots of the cubic equation fi - ISt^ + 71t - 105 = 0, and are therefore equal to 3, 5, 7. 142. When a; = a in the given equation, y = b + c-am the transformed equation. Now b + c-a=a + b + c-2a= -q-2a. If therefore we put y= -(q + 2x) we have only to eliminate x between this and the given equation. Now 8i^ + 8qx^ + 8r=0, a,n6i -2x=y + q. Hence we have (2 + 2/)3-2j(g + i/)2 + 8r=0. 143. (1) We have xS= nx + {n-l)x^+{n-2)x^+...+2x"-'^ + x"; S = n+{n-l)x + {n-2)x^ + {n-S)3^+...+ ic"-'; hence (x-l) S= ~n + {x + x^ + x^ + ...+ x"-'^+x"). (2) Let the scale of relation hel-px- qx^ - rx^ ; then S = 3- a;-2a;2-16a;3-28a;*-676a;5-... -px8= - 3px+px^ + 2px^ + 16px* + 2Spx? + ... -qx^S = -3qx' + qx^ + 2qx^ + lijqx^ + ... -rx^S = -3rx^ + rx*+2rx^ + ... Thus S(l-px- ■qii?-ri?) = 3-{3p + l)x-{3q-p + 2)x''; where 2p + q-3r=U; 16p + 2g + r=:28; 28p+16g' + 2r=676; whence p=-5, q = 50, r=8. Hence 3 + 14a:-157a;2 l + 6»-50a;=i-8a!3' 504.] MISCELLANEOUS EXAMPLES. 307 (3) Applying the method of difEerenoes, we have 6 9 14 23 40 8 5 9 17 2 4 8 Hence as in Art. 401 we may assume «„ = a . 2"~^ + hn + c; and we have a + 6 + c = 6, 2a + 26 + c = 9, 4o + 34 + c = 14; whence a = 2, 6=1, c = 3. Thus M„=2»+n + 3; and S„=(2''+i-2) + ^«(n + l) + 3n. 144. We have from the first equation a(yz + zx + xy) = xyz; also from the second and third equations, 2{yz + zx + xy) = {x + y+z)^-(x^ + y^ + z^) = V-c^. Now cP=a? + y^ + z^ = (x + y + z)'^-S{x + y + z) {yz + zx + xy) + 3xyz = b^-S{yz+zx+xy) (b-a); .: 2i' = 263-3(62_c2)(6-a). Again, b^-cP = Z {(x + y + z) {yz + zx+xy)-xyz} = S{y+z){z + x){x + y); which by hypothesis is not zero. Hence b cannot be equal to d. 145. The first derived function of 3a!* + IBx^ + 24a;' - 16 is 12a; (^2 + 4a; +4); and the H.C.F. of these two expressions is x^ + ix + 4:; hence the first expression contains the factor (x+2)3; the remaining factor is 3a! -2. Thus Q the roots are - 2, - 2, -2,-. 146. From the data, we see that the sum of n terms of the series 1, 3, 5, 7, . . . is equal to the sum of m - 3 terms of the series 12, 13, 14, ... ; hence m2=-^ {24 + (n-4)}; that is, rfi-nn + 6Q = 0; so that n=5 or 12. ,^r, -nr u 1111 147. We have x=__^__; that IS, x==^ -; 10a; + 7 hence Sa:^ + 2a; - 1 = 0, and x='^ — . 5 20—2 308 MISCELLANEOUS EXAMPLES. [PAGE 148. The given equation may be written (x + af-3bc (x + a) + b^ + c^=0, or y^-3bcy + h^ + c'=0,y/heiey=x + a. By putting y=s + t, and proceeding as in Art. 576, we have st = bc, and s^ + 1?= -{b^ + c^); whence s'= -6^ and f'= -c'. Hence the values of 2/ are -(6 + c), -{ab + or'c), -{u^b + uc). 149. By Art. 422, it foUows that a"-a = M{n), and b" - b = M (n) ; hence {a" + b") - (a + b) = M (n) ; and therefore by dividing each side by a + 6, we have o''-^-a''~'-'6 + a"-*6^-... -a6"-^ + b"-'-l=M(n), since a+b is prime to n. But by Fermat's theorem a"-^-l = itf (m), and b'^~^-l=M (n); hence a"-'b- a''-^b^+ ... + ab"-'=M {n) + l. 150. The generating function 1 - nbx- 1 I n 6 ~ (1 - axy^ (1 - bxf ~ a-bl{l- ax)'^ (1 - hxf) ' Therefore {a -b)u„= na''x''~i - n6»a;»-i. The sum of the series a + 2a% + 3a^x^ + ... + 7ia"x''-^ is easily found to be a - nn"+' a:" a^x (1 - a"-! x"-') 1-ax (1 - nx)^ 151. Here a^ + b^ + c''= -2p, since a + 64-c = 0, and bc + ca + ab=p. „,, • ^v • i- ft^ + f- a'' + 6'' + c2 2p When X = a in the given equation, y = = a = - — - a o a o in the transformed equation. We have therefore to eliminate x between x^+px + q = and y= -~-x, or x^ + xy + 2p = 0. Eliminating t' we have x^y +px-q = 0. From the last two equations we obtain x^ _ X 1 - qy-2p^ ~ ipy + q~ p-y'^ ' .: {2py + qr=(y^-p){qy + 2p''). 152. If a + b + c=0, we have {a + b + cf=0, that is a'' + b' + c^= -2(bc + ca + ab); hence a*+b*+c* + 2b^c^+2c''a^ + 2a%^ = i(bV + c'w' + a%'') + Sabc(a + b + c); thus a* + b* + c*=2{bV + c'a' + a'b''). .: {a' + b-'' + cY = a'^ + ¥ + c^ + 2(b'c^ + c''a'^ + a-¥) = 2{a*+b* + c*). [Compare XXXIV. b. 11.] 505.] MISCELLANEOUS EXAMPLES. 309 Put a=y + z-2x, b = z + x-2y, c = x + y-2z; then a' + b" + c' =6 {x^ + y' + z' - yz - zx - xy) ; and the required result follows at once. 153. (1) Proceeding as in Art. 576, we have f + z^= -133, and yz = 10; whence we obtain y^= - 125, z'= - 8. Thus the real root is - 5 - 2 or - 7 ; and one of the imaginary roots is -50,-20^=2-3.. = ^"^^^"^ (2) The sum of the roots is a-a + b-b + c=c; but the sum is also 4 ; hence c = 4. Removing the factor x - 4 corresponding to this root, we have x* - lOi^ + 9 = 0; that is, {x^ - 9) (x'^ - 1) = 0. 154. Let Q be the quantity of work done by the man in an hour; P his pay in shillings per hour ; and H the number of hours he works per day ; then by the question Qa -r— = — — , where m is some constant. W Let W represent the whole work ; then in the first case he does ^rr per W mxl hour: hence =-7= — ttt- 54 ;^9 Let X be the required number of days ; in this case he takes 16x hours to , ^, , , W mxli do the work; hence rs-= — ttt- ^ lex ,^lb , T . . 16x 1x4x2 , „ by division, -z-r = — = — =— ; whence x = 3. 54 3 X 3 155. From Art. 383, we have s„= = n (n+ 1) (n + 2) ; o 1 1 and from Art. 386, we have ff'„_i = Ta 18 3n(n + l)(n + 2) Hence -4a6c(a + 5 + c) = 0; that is, fc''c2 + cW + a^ja _ 2a26c - 2a62c - laW^ = ; or (6c + ca-a6)2 = 4a26c; hence bc + ca-ab= ^ija^bc; that is, Jbc=i^iJca=±Jab; or -j- ± -7- ± _ =0. ,Ja Jb ^c 164. (1) Then«''term=n(n + l)(7i + 3) =n(n + l)(n + 2] + n{n + l); hence S = ^n(n+l){n+i)(n + %) + -n(n+\)[n + 2) [Art. 383] = in(» + l)(n + 2)(3n + 13). 507.] MISCELLANEOUS EXAMPLES. (2) The m"" term = ; ^ _ {n + l){n+2)-3{n+2)+4 : |m + 2 _ J_ 3_ 4 "" In (to+1 Ito + 2' Hence the Bum = («-l)-3(e-2)+4^e-2-i^=2e-5. 165. By Art. 253, we have ^±|i^>(6cdj*; that is «-o>3(6cd)*. Similarly, s-b>3 (cda)i ; s-c>3(da6)4; s-d>3 (aSc)*. By multiplying together all these inequalities we have the required result. 166. (1) We have 2ijx + a = 2 Jy-a + 5ija; and i Jx-a=2jy + a + i^Ja, by squaring and subtracting, we have 8a= -8a + 16a + 20;^a Jy-a-lija Jy + a; whence 5 Jy -a = 'A Jy + a; and ?/= — . Substituting for y, we have 2 Jx + a= I 5 + -75 J ija; ,, ,. , 59 + 30^2 51 + 30^2 that IS, x+a= . ^ n; or a;= a "" (2) We have 2 (2/2 + 2a; + x2/) = (x + j/ + 2)''-(x= + 2/2 + 22); whence yz + zx + xy = 3. Again x^+y'' + z^-yz-zx-xy = 0; and therefore x^+y^ + z^-3xyz = 0, so that 31^2=6, and xi/2=2. Now a;, j/, 2 are the roots of the equation fi-(x + y + z)t^ + (yz + zx + xy) t-xyz—0. 314 MISCELLANEOUS EXAMPLES. [page Thus X, y, z are the roots of the equation t^ -Zl^ + it-1 = ^, or (t-2)(t2-t + l) = 0; whence f = 2, or 167. From these equations we have X y z z X y y z X 1 y 1 X 1 z that is, l{x^ + y^ + z''-3xyz) = x'' + y^ + z^-yz-zx-xy, or Z(a; + ?/+2) = l. 1 Thus and therefore l=m=n=- x + y+z 3k^={x + y + z)K 168. It is easily seen that the numerator vanishes for each of the values a = 0, 6 = 0, c=0. Hence it must be of the form kabc where fc is a numerical quantity. Similarly the denominator is of the form labc; hence the value of the fraction is some constant quantity m. To find m put a=6 = c = l; 3 + 1 then m = - — = =2. 169. 'Pat X=x^-yz, Y=y^-zx, Z=z^-xy; then the given expression is the product of the three factors X+Y+Z, X + aY+io^Z, X + w^Y+uZ. Now X+Y+Z = x' + y^ + z^-yz-zx-xy = {x + ST > t^^^* '^i '° ^> 6> ^ respectively. itJi 11 o^ Suppose then that he walks, drives, and rides 3ft, 6fc, 8A; miles in the hour; then if x, y, z be the distances AB, BG, GA, wo have '^ + 2' 4. ^ -151 ■ik 508.] MISCELLANEOUS EXAMPLES. 315 Again since he walks, drives, and rides 1 mile in half an hour, vre have 1111^ ,5 Thus 8x + iy + 3z=i&5, 4a! + 3t/ + 8z = 360, a!+2/ + z=82Jj whence a = 37 J, y = SO, 2=15. 171. The expression = n {{«,«- 8)- Iv? (n" - 2) } = 7i(«2_2)(n*-5«2 + 4) = «(ni-2)(n-2)(n-l){n + l)(7i + 2); and is therefore divisible by 15 or 120. If n is a multiple of 7, the given expression is obviously divisible by 7. If n is not a multiple of 7, then rfi-l = M(l), by Fermat's theorem ; and therefore »i^-8 = iK (7). In this case the expression is also divisible by 7; thus it is divisible by 7 x 120 or 840. 172. (1) Substitute ^ = 23- a: in the first equation; thus Va;^- 12a; + 276+ >/a:2-34x + 529 = 33. But (x« - 12a; + 276) - (a;^ - 34a; + 529) = 11 (2a; - 23) identically; hence ^a;2-12x + 276 - v'a;'' - 34a; + 529 = ^^111^ . o By addition, 2 Jx^ -12a; + 276= -4^ ; whence 9 (a;2-12x + 276) = (x + .38)2; that is, a;'^- 23a; + 130=0, and therefore x = 13 or 10. (2) From the given equations we have u a y c z b X d " _ ex bd ae ad Hence V = -r. « = — . «= — = — • " d' X y X Substitute in the last equation ; thus , d{a-b) whence x = -^ . d-c 173. We have (^+2,+,+...)g+i+i+...)=„+g+g + g+£)+...; where n is the number of the quantities x, y, is, ... 316 MISCELLANEOUS EXAMPLES. [PAGE On the right side, each of the expressions within brackets is greater than Iai lift —.111 2x -i-g — '-'t ; that is, greater than v?. Thug (x + y + z + . ..){- + - + -+.. ]>n^. „ , s-a s-b s — c Put x= , y = .2 = ,...; s " 5 s then x + y + z+ ... = =n-l; s whence on substitution we have the required result. 174. Suppose that he bought x cwt. of cotton, and exchanged each cwt. for y gallons of oil, and sold each gallon for z shillings ; then he obtained xyz shillings ; hence {X + l)(y + 1) (2 + 1) =xyz + 10169 ; {X -l){y- 1) {z - 1) =xyz - 9673 ; and X, y, z are in G.P. so that xz—y^. From the first two equations, (yz + zx + xy) + (x + y + z) = Wl&%; (yz + zx + xy)-(x + y + z) = ^(:12; whence yz+zx + xy = 9^20, and x + y + z^iiS. But yz + zx^xy=yz + y'^ + xy=y {x+y + z); hence 2iSy = 9920 ; that is, 2/ = 40. Thus x + 2 = 208, and 2:2 = 1600; whence a; = 200, 2=8. 175. The expression vanishes when x=a, when x = 'b, and when x=c; and is therefore divisible by (x ~a)(x- b) {x - c). Again, the expression is of 5 dimensions in x, but the coef&cients of both a;' and x* are zero; for the coefficient of x^= -(6-c) - (c-a)-(a-6) = 0; and the coefficient of x^ = S{a(6-(;) + 4(6-c)(!) + c-a)}=0. Hence the given expression = / (a, 6, c){x-a) (x-b) (x-c), where /(a, b, c) is a function of a, b, c of three dimensions. Now the given expression vanishes when b=c, e = a, a=b; therefore it must be of the form k(b-c)(c-a){a~b){x-a){x- b) {x - c), where h is constant. By putting x=0, we have Sa(6-c) {b + c-a)*= - kabc {b - e) (c - a) (a-b). Finally putting a =3, b = 2, c = l, we have 12i=3 .0*-4 . 2*+4»=192; whence A =16. 508.] MISCELLANEOUS EXAMPLES. 317 3-\-y oi-\-B-\-y j) 176. Putting y = - — - , we have «= — - — - - 1 = - - 1 j hence if x has a a a , V P 1 V + 1 the value a, then y has the value - - 1 ; so that y = - - 1, or - = . " X " X X p Substituting in the equation -5 - -+ 1 = 0, we have 177. We have {a? + 62) (c^ + d?) = a^'' + }fid? + h'^c'' + a'A^ = (ac±6d)« + (6cTad)2 (1). Write ^ for ac ± hd and B for 6c =f ad ; then (a=+62) (c2 + )2+(BC'=F^B)2 =p^ + q'' say. It is clear that each pair of values of A, B, with each pair of values of C, D gives us two pairs of values for p, q ; thus we have in all eight solu- tions. One of these, namely that obtained by taking the upper sign throughout, is p=AG+BD = [ac + lcl) {eg +fh) + (be - aO) (fg-eh), q=BG - AD = {bc -ad) {eg +fh)- (ac + bd) {fg - eh). [This solution is due to Professor Steggall.] 178. We have {x-y){x^ + xy + y^) = 91 ; that iB{x-y) (61 + xy) = 91. Also {x-j/)2 + 2a;!/ = 61; hence by putting x- 3/ = u, and a;^=j;, we obtain u(61 + i;) = 91, and m2 + 2« = 61. Multiply the first equation by 2, and substitute for v ; thus m(183-«2) = 182; or tt*-183«+ 182 = 0. 318 MISCELLANEOUS EXAMPLES. [PAGE Hence (u -l)(u' + u- 182) = 0, or «= 1, 13, - 14 ; 135 and therefore « = 30, - 54, — =- . Thus we have K- J/ = 1,1 x-y = 13, ] x-y=- 14,^ a;M = 30f' a;w=-54J' 135 V ' 179. The number of ways is equal to the coefficient of a^™ in the ex- pansion of (a;" + xi + x^ + . . . + X™)*. This expression =( — 1 = (1 - x^+'j* (1 - x)-*. Thus we have to find the coefficient of x^™ in (1 - 4x™+i) (1 - x)"*. The coefficient of x" in (1 - x)-* is ('"+1) ('" + 2) (r + S) . hence the required coefficient = g(2m-l-l)(2m + 2)(2m + 3)-gm(m+l)(m + 2) = -(m+l)(2m= + 4m-|-3). 180. Wehave 0(3=1, a + |8=-p; 78=1, 7 + 5= -3; hence (, 186. (1) We have 2{yz + zx + xy) = {x+y + z)''-x^-y^-z''=i; that is, yz + zx + xy = 2. Again, {x+y + z)^='Sx^ + 32x^ + Gxyz; that is, SXx''y+6xyz = S + l = 9. 320 MISCELLANEOUS EXAMPLES. [PAGE But ^xh/ = 'Zx:Sy2-3xyz = 4:-3xys; hence 12 - 3xyz = 9, so that xyz = l. Now x, i/, z are the roots of t^-(x+y + z)t^ + (yz + zx + xy)t-xyz=0; thatis.of f'-2«2 + 2s-l = 0, or (t-l)(t^-f + l) = 0. (2) We have (x-y+z)(x + y-z)=a^; (-x + y + z)(x + y-z) = i^; (x — y + z) ( - x + y + z) = c^. Multiplying together the second and third equations, and dividing by the first, we have (-x + y + zY= —^ . Hence he ca ah -x + y + z=±— , x-y + z= J=-j- , x + y-z=± — . From the given equations we see that these results are to be taken all with the positive or all with the negative sign. 187. Let X denote the numbEr of Scotch Conservatives, and therefore the number of Welsh Liberals. The number of Scotch Liberals is therefore 60 -a;; hence the Scotch Liberal majority is 60 -2a;, and therefore the number of Welsh Conservatives is 30 -x; hence the number of Welsh members is 30. The Irish Liberal maiority=-(60-2a;) = 90-3a;. We may then represent the number of members by the following table Conservatives Liberals English y, z; Scotch X, 60 - a; ; Welsh 30 -X, x; Irish u, M + 90-3X. Thus we have the following equations : z + u-Zx + 150 = y + 15; that is, 3x + y-z-u=lZ5; 2/ + « + 30 = 2« + 5; that is, y-'iz + u^ -25; y-i; = 2M + 90-3x + 10; that is, Zx + y -z-2u=\m; y + « + 60 + 30 + 2u + 90-3a:=652; that is, -3x + y+z + 2u = i72. From the first and third equations, we have «=35 ; hence Bx + y-z = nO, y-2z=-60, -3x + y + z = i02. Adding together the first and last of these equations, we have 2j/ = 572 or y=28&; hence ^ = 173; anda; = 19. 188. It is easy to prove that the expression on the left contains the factor (6 -c) {c- a) {a-b); the remaining factor being of thi-ee dimensions and symmetrical in a, h, c must be of the form k'Za^ + l2,a''b+mabc, where k, I, and m are numerical. A comparison of the terms involving a' shews that 4=1. 510.] MISCELLANEOUS EXAMPLES. 321 Again there is no term involving a* on the left; while on the right these terms arise from (b-c) { -a'' + a(b + c)-bc} {ka.'>+la^ib + c)+ ..A; hence k{b + c)-l(b + c)=0; wh.enoel=k=l. To find m, put o=2, 5=1, c= - 1 in the identity a" (c - 6) + 65 (a - c) + c5 (6 - a) = (6 - c) (c - a) (a - 6) (iSaS + ZSa^i + maftc) ; thus 2»(-2) + 16(3)-15(-l) = (2)(-3)(l)(8ifc+4i-2m); that is 10 = 8A; + 4i - 2m ; whence m = 1. 189. Keeping the lowest row unaltered, we see that the determinant a3-l a^-l a -1 1 = {o-l)» 3a' -3 3a- 3 a^ + 2a-3 2a-2 2a -2 a-1 = (a-l)= a + 2 3 1 1 1 2 a^-l 8a2-3 3a-3 a2_l a2 + 2a-3 2a-2 a -1 2a-2 a-1 a^ + a+l 3a + 8 a+1 a+3 1 2 = (a-l)3 = (a-l)5| a + 2 8 l=(a-l)8. 11 a2 + a-2 3a-8 a-1 a-1 1 2 1 190. We have ^5+f + a + c-26 =0; ac ' ac-b (a + c) + l that is, 6'i(a + c)-6{(a + c)2+2ac}+2ac(a + c) = 0; or {6(a + c)-2ac} {6-(a+c)}=0; hence b = a+c, or 6 (a + c)-2a<;=0. 191. (1) Denote the roots by a, a + 2, 11 -2a, the sum of these roots being 13. Since the sum of the products of the roots two at a time is 15, we have a^ + 2a + (2a + 2) (11 -2a) = 15; that is, 3a' - 20a - 7 = 0, or (8a + 1) (a - 7) = 0. Again o (a + 2) (11- 2a) = -189; this equation is satisfied by a=7, but not by a= - - . Thus the roots are 7, 9, - 3. (2) The equation whose roots are 2 ± ij^ is a;^ - 4a; + 7 = 0. Now a;*-4a:2 + 8x + 85 = (x2-4x + 7)(a:=' + 4x + 5); hence the other roots are given by x' + ix + 5=0. 192. We have thus H. A. K. a-b . ^ a-b a'bVlbc + ca + ab); or tt-„ + -^„+ -wr,>bc + ca + ab. b'c' c'a' a'b^ „ fa? Vy . .^. , a« 6= 20=62 ^°^ [be - ra) '= p°''*^^" -' ^"""^ W^M'^^^- „, «« 6« c« a^b^ bh^ c^a? Again, r- XB positive; hence —^^ + ,„->2c^ \a J a^ o" _,, 6V c^a^ a^ft" „ ,„ Thus — 2- + -j^+ -5->a= + 62 + c2. a'' 0'' c^ Finally it is well known that a?+b'' + c^>bc + ca + ab. Hence a fortiori the required result is true. 200. Let X be the time in hours after which B dismounts ; then A has gone ux miles, and B and G have each gone vx. Now B continues to walk hours; therefore the whole time occupied is xH , for his wo^fcire;; pace is the same as A's. When C starts back to meet A they are (b-«)x miles apart ; therefore if they meet in p hours we have / \ , \ 1 {v-u)x p(u+v) = (v-u)x; whence p=^ . ' u + v Again they meet (x+p)u miles from the starting point, so that the dis- tance remaining is a-(x+p)v, miles, and the time occapied in driving this ■ tl — (x + p) w , distance is ^^— hours. V Now the number of hours after B dismounts a-vx a-(x+p)u = =p + i ^^— . U V From this equation we obtain a-x{u + v) x{u + v) + up = a, 01 p = - u By equating this to the former value found for p, we obtain x= -^ — =— ^ . v' + 3uv Hence the whole time occupied = xH = - . hours. U V 6U + V 201. We may represent the city by a rectangle whose sides are a and 6. Let a, running N. and S., be vertical, and 6, running E. and W., be hori- zontal. Then it is clear that whatever route is chosen the man has to travel a distance equal to a in the vertical directioa and a distance equal to b ia 326 MISCELLANEOUS EXAMPLES. [PAGE the horizontal direction. Now 6 is the aggregate of m-1 horizontal dis- tances, and a is the aggregate of n-1 vertical distances, and the ■m+n-2 portions which make his whole path may occur in any order. Thus the number of ways is equal to the number of permutations of m+n-2 things m - 1 of which are of one kind and n-1 are of another kind. 202. Put u for i,'x + 27, and v for i]55-x; then m* + j)<=82, and u + v = i. Eaise both sides of the equation to the 4"' power ; then we have 82 + iim (m2 + 2uv + v^) - 2uV =256, that is, 82 + 64«t) - 2mV= 256 ; or «V-32Mt)+87 = 0; whence uv = 29, or 3. Also u-|-D = 4; .-. M=2±5,yn, or 3, or 1; .-. x + 27 = {2J.Sj^^y, or 81, or 1. 203. If Sjn denotes the sum of 2n terms of the series ab + {a + x){b + x) + {a + 2x)[b+2x) + ...; S2„=2nab + x(a + b){l + 2 + 3 + ... to 2n-l terms) + a;2(12 + 22 + 32+... to 2n-l terms) = 2nab + n(2n-l){a + b)x + ^n{2n-l){in-l)x\ By writing n for 2m, we have S^=nab + ^n{n~l){a + b)x + ^n{n-l)(2n-l)x^. :. S,^-2S„=n^(a + b)x + 'n?{2n-l)x''='n?x{a + b + {2n-l)x}. If I is the last term, l-ab = {a+2n-l .x){b + 2n-l. x)-ab = {2n-l)x{a + b + {2n-l)x}. .: S2„-2S„ : l-ab=rv':2n-l; which proves the proposition, since Sj^i-iS^ or {S2n-SJ-S„ denotes the excess of the last n terms over the first n terms. 204. (1) Let ^ be the n* convergent; then p„=^Pn-i-Pn-ii ^° ^^^^ the numerators of the successive convergents form a recurring series, whose scale of relation is 1 - 2a; + x^. 512.] MISCELLANEOUS EXAMPLES. 327 Put 8=pi+p^x+psx''+...; then, as in Art. 325, we have S=^i±^^^^- . But ^1 = 1, P2=2; hence S= .-z rj, and p^^n. (l — x) Similarly if S'=qi + q^ + q^^+ ..., we shaU find q. = n+l. Thua ^ = -^ . (2) The scale of relation is 1 - 3a; - ix^. With the same notation as in the preceding case, we have i'l+Pa^+l'a^ +...- i_3^_4^. - l-3x-4a;2-5 U-4x + l + a;J " a+„ = I i4" + ( - 1)"-'} ; and ?„=! {4"+i + ( - 1)"}. 205. Put (o-«)(2/-2) = o, {a-y)(z-x)=^, {a-z){x-y) = y; then after transposition we have to prove that is zero. Now this last expression has a factor + /3 + 7 which from the above values is evidently equal to zero. 206. The expression whose value is required can be written in the form -[[n-m0){n-my)(n + ma) + ... + ...] (n - ma) (n - m^) [n - my) The numerator = - [re^ - Ti'm (a - /J - 7) + nm^ (§y-ya-a§)+ m^apy + two similar expressions] = _ [3nS - n%S (o - ^ - 7) + nm^S (^-y - 7" - a/3) + 3m'a;87] = -3'n?-'rAn{a+p + y) + nm^ffiy + ya + ap)-3nv'a§y = - 3n' + mn'q + 3rm', by the properties of the roots of the equation. The denominator =n'- ■hhn, (o + 18 + 7) + nm^ {fiy H- 7a + ojS) - m'a/37 = n' + nm^q + irm?, 3rm? + nm^q - 3n' the required expression = rm^+nmi'q + n' 328 MISCELLAJfEOUS EXAMPLES. [PAGE 207. If X is the population at tlie beginning of the year, then the popu- X X 1531j; lation at the end of the year is a; + j^^r - -77; = -^^ jr ; hence if n be the re- do 4d I0J.0 quired number of years, ( — — 1 x=2x; that is, n (log 1531 -log 1518)= log 2; or ■0037034re= -3010300, and n=81. 208. We have {l-x^)^=(l-x)''(l + x + xy={l-x)''[ao + a^x + a^x^+...). Equate the ooefScients of s'; then if r is not a multiple of 3 the coefiBoient of x'' on the left side is zero, and the required result follows immediately. If r is a multiple of 3 it is of the form 3m, and on the left the coefficient ofics^is (-ir I,"' M , or (-1)"3 "' m 1 (» - m) I ' i' (-1)' 209. Denote the number of Poles, Turks, Greeks, Germans, and Italians hj X, y, z, u,v respectively ; then we have 1 , 1 „ x = -^u-l=^v-3; y + u-z-v=3; z + u=-^{x+y + z + u + v)-l; z + v = Ys{x + y + z + u+v). From the fourth equation, we have x + y-z-u + v = 2; subtracting this from the third equation, we get 2u-2v-x = l. But from the first two equations, m=B!.-2 ^i!^l)a;n-2+(_ l)«-l„itn-l+ (_ l)na;«. 1 • ^ Multiply these two results together, and equate the ooeffioients of x"-^. Then, if S stand for the left-hand memher of the proposed identity, we have (-l)"-'S = the coefficient of x'^^ in the expansion of (1-a;)-'. .-. {-l)''-iS = l = (-l)2», that is, S = (-l)'»+i. 212. (1) If we form the successive orders of difEerences, we obtain 6, 24, 60, 120, 210, 336,... 18, 36, 60, 90, 126,... 18, 24, 30, 36,... • 6, 6, 6,... „ « io, ■,^ 18(ra-l)(7i-2) 6(«-l)(7i-2)(n-B) Hence it„=6 + 18(ra-l) + — ^ r^ '-+— '\„ ■' [2 [b =n(n + l)(n + 2). [See Art. 396.] ••• S„=jn(ra+l)(n + 2)(m-l-3). (2) Here u^={n + lf{- x)"--^ ; and therefore the series is recurring and (1 + x)^ is the scale of relation. [Art. 398.] Let S=4- 9a; + 16x2- 25x3 + 36x«-...; then 3xS= 12x -27x2 + 48x3 - TSx* +... ; 3x^5= 12x2-27x= + 48x*-...; x'S= 4x3- 9a:*+.... By addition, (1 + x)3S= 4 + 3x + x=. (3) Putx=|; then S = l . 3x + 3 . Sx' + S . 7xS + 7 . 9x*+ ...; thus u =(2ra-l)(2ra + l)x". Hence the scale of relation is (1-x)'. [Art. 398.] 3 + 6ic 3)^ Proceeding as in (2), we shall find S = —p: — —5- = 46. 330 MISCELLANEOUS EXAMPLES. [PAGE 213. Add together the first and third rows, and from the sum subtract twice the second row; also subtract twice the first row from the third row; thus 4a! 6i + 2 8a! + l =0; -3 -4 3 1-4 hence 4x(12)-(6a; + 2) (-3) + (8a; + l)16=0; and therefore 194a; + 22=0 ; that is, x= --=. 214. (1) This follows by adding together the inequalities a« + 6V>2a6c; 62 + cV>2a6c; 2abe. (2) The two quantities aP-b" and a^-b^ are both positive, or both negative; hence (a» - 6") (a«- 6«) is positive; that is, aJ>+i + iP+Q > aPb" + a%i>. Similarly, a^^i + c»+8 > oPc' + a«cP ; and jp+a + ^p+a > jPc* + bid" ; and so on, the number of inequalities being J re (n - 1). By addition, (?i-l)(aJ>+«+6i^+«+c'^«+ ...) > SaPJi; hence m(a''+«+6i'+«+c»+«+...) > So'^s+Sai'fi'; which proves the proposition. 215. The given equations may be written (y-a)(z-a) = a^ + a; (2-o)(x-a) = o^ + /3; (x - a)[y - a) = a? + y. Hence (x -a){y-a){z-a)=± {{a'^ + a) (a^ +/3) {a^+y^i. Divide this result in succession by each of the given equations. 216. The given expression = {1.2»-i + 2.3»-i+... + (re-2)(«-l)'>-i} + (re-l)TC»-' + n-l. Now by Fermat's theorem each of the expressions 2"~i, 3"-', ... (n - !)"''■ is of the form 1 + M{n). :. the given expression = {l + 2 + 3 + ... + (n-2)} + (?i-l) + (n-l)7i»-i + lf(n) which is a multiple of n, since — Lj — - is integral. 513.] MISCELLANEOUS EXAMPLES. 331 217. The number of ways of making 30 in 7 shots is the ooefacient of a;'" in (x" + x'-' + x'^ + se' + x^Y ; for this coefficient arises out of the different ways in which 7 of the indices 0, 2, 3, 4, 5 combine to make 30. Now (xi' + a^ + s^ + x2 + l)'={x4(x + l) + !c' + x2+l}7 =x^ {x + ly + 7x^ (x + 1)6 (i3 + a;2 + 1) + 21x20 (x + 1)« (x3 + x2 + 1)2 + 35x16 (a + 1)< (x' + x= + 1)3 + . . . = 21 + 7(l + 15 + 20) + 21(2 + 5) = 21 + 252 + 147=420. 218. Denote the complete square by (x + Si:)^; then since the coefficient of X* in the given expression is 0, the complete cube will be (x - 2/[)*; thus x5 - 6x3 + cx2 + (ix - e = (x + 3fc)2 (x - 2*:)s= x6 - 15 fc^x' + lOfe^ + 60ft*x - 72P. Hence by equating coefficients, we have 15k^=b, 10k^=e, 6Qk*=d, 72k' = e; thus 36J;2= -— = — = — = —. 5 6 c e^ 219. There are four cases to consider for the bag may contain 8 white, 4 white, 5 white, or 6 white balls, and we consider all these to be equally likely. 3 2 1 __£ 3 2 _5^ j4 3 _^ 5_ j£. ^^~9" 8 "7' ■''^"lO" 9 "8' ''3~11'10 "9' ■''*~12 ' 11' 10' thatis, p^=^, p,=^, 1.3=33, ^*=n- . ^_ «2._^-_?4=JL ■■55 154 280 420 909' The chance of drawing a black ball next 6 3 2 _ 55 6 154 3 280 2 420_677 ~ 909 "*" 7 '^ 909 "^ 4 ^ 909 "*" 3 '^ 909 ~ 909 ■ 220. Here 2S = {V + 2^ + 3^+ ...+nY-{i-* + ^*+3* + -+n% Now by Art. 405, _n^ n* rfi n ~ F "^ l" "•" 3 ~ 30 = "f:(n + l){2n + 1) (3n2 + 3n - 1). 332 MISCELLANEOUS EXAMPLES. [PAGE Thus 25= -^ >^^ >- - -^ 3^ = -i-m(n + l)(2re+l){5n(2n2 + 3n + l)-6(3n2+3n-l)} = ^n{n+l)(2n + l)(n-l)i2n-l)(5n + e). 221. On reduotioi, we obtain a;2{a2 (6 -c) + ... + ...}- a; {o2(62_c2) + ... + ...} + {a25c(6-c) + ... + ...}=0; if tlie roots of this equation are ec[ual, we must have {a2(&2-c2) + ... + ...}2-4{a2(6-c) + ... + ...}{a'6c(6-c) + ... + ...} = 0. The coefficient ot a* in the expression on the left = (j2_c2)2-46c(6-c)2=(6-(;)2{(6 + c)2-46c} = (6-c)*. The coefficient of fi'y" = 2 (c2 - o2) (a2 - 62) - 4a5 {c-a){a-b)-ica{c-a){a-h) = 2 (c - a){a-h) {{c + a){a + b) - 2ab -2ca}=-2{c- a)^ {a - b)K Hence the condition reduces to a*[b-cy + p*{c-a)* + y*{a-b)*-2p^r^(c-a)->{a-by -27V {a- 6)2(6 -c)2- 20^^/32(6-0)2(0-0)2=0. But the expression on the left is now of the form a;< + y* + 2' - 22/2«2 - 22 V - 2x^^, the factors of which are -{x + y + z){-x + y + z){x-y + z){x+y-z); and this expression vanishes if a;±^±z=0. 222. Here 2»-i, (n-2)2''-3, ^"~^^ ^"~^^ 2"-°,... are the coefBoients of a;"-i, x"-', a:"-'*,... in the expansions of {l-2x)-^, (l-2a;)-2, {l-2x)~',-.. respectively. Henoe the sum required is equal to the coefdoient of a;""^ in the expansion of 1 a" a:» l-2x (1 - 2a;)2 ■*■ (1 - 2a;)3 "• and this may be regarded as an infinite series without affecting the result we wish to prove. But this series is a G.P. whose sum ~l-2a: ■ \ "'■l-2a;j~(l-a -X)' Therefore by equating coefBoients of a;"-' we obtain the result stated. 514.] MISCELLANEOUS EXAMPLES. 333 223. (1) By addition, we have (x + j/ + z)2=225; that is, a; + 3/+z= ±15. Again x'-y''-2z{x-y) = 0; that is, (x-y){x + y-2z) = 0; whence x=y, or x + y=z2z. It x=y, we get, by equating the second and third of the given expressions, x''-2xz + i?= -3, or x-z= ±^-3. Combining this with 2x + z= ±15, we have !c=j/ = i(±15±733), z=i(±15T2x/^). If x + y-2z=0, we have by combination with a + ?/ + «= ± 15, the equa- tions 2=±5,x + j/=±10. Substituting in y' + 2zx = 76, we have 2/»±10{±10-2/) = 76; that is 2/2 ± 10?/ + 24 = ; whence ?/= ±4, ±6. (2) Put x = a + h, y = b + k, z = c + l; then from the first two equa- tions, h + k+l=0, h + h + i=Q. a b c , h k I whence —pr r = rr-. 7- = — = X say. a{b-c) b{c-a) c{a-b) •' From the third equation, ah + bJc + cl = bc + ca + ab-a''-b''- c'; thus A {a2(6 -c) + b^(c-a) + c^{a-b)} = bc + ca + ab-a^-b'-c^; a^+b^ + c'-bc-ca-ab \=- ib-cj{c-a)(a-b) 224. Let the points in one line be denoted by Jj , A^, Ag, ..., A„, and those in the other line by B^, B^, ..., B^; and let Aj^, £j be towards the same parts. Then from a diagram it will be seen that A^Bi will cut m - 1 lines diverging from A^ ; ^JBi 2(m-l) A^U^; ABi 3(m-l) ^1,^2,^3; A^i (n-l)(TO-l) A^, A^, ..., A„. Again, ^2^2 '^^ ^^^ m-2 lines diverging from A^ ; ^3^2 2(m-2) A^,Ai; ^A 3(m-2) A^,A^,A,; ^^2 (n-l)(m-2) A^,A^, ..., A^. And so on, taking all the m points B^, B^.-.B^ in succession. 334 MISCELLANEOUS EXAMPLES. [PAGE Finally, /^jB^.j will out 1 line from J^; ^jB^-i 2 lines diverging from Jj, jIj; ■^4-^m— 1 ^ -^l* ^2' -^3* A-Sm-1 n-1 Ai, A^, ..., An. We have now enumerated all the points; for A^B^ outs none of the lines from Ai, A^B^ cuts none of the lines from A^, A^, and so on. The number of points we have indicated is clearly equal to {l + 2 + 3 + ... + (n-l)}{(m-l) + (m- 2) + . .. + !}, which IS equal to —^- — - x — ^ — - . 225. We have for it is obvious that the coefficient of y is 1. Equating coefficients of powers of a, we have a + l = 0, or o= -1; 6 + 2a = 0; that is, 6=-2a=2; c + 36 + a=0; that is, c= -5; d+4c + 36 + l = 0; thatis, d=13; whence aH - Sabc + 263= 13 - 30 + 16 = - 1_ 226. Denote the price of a calf, pig, and sheep by x, x - 1, a: - 2 pounds respectively ; and suppose that he spent y pounds over each of the different kinds ; then we have the equations ^ + _^ + _V_ 47, and^-^ = -9 . X x-1 x-2 x-1 X x-2 9x (x — 1) From the second equation y = — _ •' ; substituting in the first equa- tion, we have 9x (x - 1) 3x°-6x + 2 x-2 ■x(x-l)(x-2)~ ' that iB, 27x'' - 54x + 18 = 47x2 - 188x + 188 ; whence (x - 5) (20x - 34) = 0, and x = 5. 514.] MISCELLANEOUS EXAMPLES. 335 227. If we put a; = l in the result of Example 2, Ait. 447, we at once obtain the desired expression for log 2. We may also proceed as follows. If = , then x_ = . Thus 1 1 1 «,^ Oj Qj a^+ a^-a^' 1_1 + I. 1 1 1 a," 1 a,' Oj Oj Ug «! a^ + X^ «! + Oj + Kg — Oj Oj + Oj - Oi + flj - flj ' 1 1 1 1 1 a 2 a 2 h = ; and so on. a^ ttg ag a^ a^ + ag ~ '^i + % — ^2 "^ ^3 By putting ai=l, 03=2, aj=3, , the theorem follows at once. 228. The number of ways required is equal to the coefficient of x^" in the expansion of (x" + a;i+x''+ ... +a""')*. This expression is equal to hence the number of ways = the coefficient of a;^*" in (1 - 6a;i»i + ISx^oa) (i _ a,)-6. \r+5 The coefficient of a;*" in (1 - x)-' is L , : thus the coefficient of x^^ is lit obtained from the product of / 143 1144 1245 \ (1 - 6x- + 15x»^) (1 + ... + |J=3 x»3 + ... + J=^ .139+ ... + J^ ,.oj . |5 [38 |i|139 |5 |2 1 229. Here 1.3.5...(4n-3)(4TO-5)a;''^\ " 2.4.6...(4ji-4)(4n-2) ' hence «„ _ 4„(4n + 2) 1_ ''""'^ «^i-(4«-3)(4»-l)-x2- Thus if a!l, divergent. If a; = 1, then Lim — !^= 1. «n+l T- / «n i\ r- n(24n-3) 8 \"n+i y {4n-3)(4»-l) 2' hence the series is convergent. [Art. 301.] [This series is the expansion of the expression in Example 105.] 336 MISCELLANEOUS EXAMPLES. [PAGE 230. Let the scale of relation hel-px- qx" ; then 288=40p + 6g, iO=ep + q; whence i) = 12, 2= -32; and the scale of relation is 1 - 12x + 32a;''. As in Art. 328 we find the generating function X - 6a;2 X ( 1 , 1 ) . ~l-r2x + 32x2 2(l-4ai l-8.<,\' and the ooefEoient of x» is ^ (4''-i + 8''-i). Therefore S„=|{l + 4 + 4!'+...+4"-i}+ | {1 + 8 + 82+.. . +8"-'} 1 4"-l 1 8''-l 2^"-' 2S"-i 5 ~2'~W~ ■*" 2 ■ "~7~ ~ ~3~ "'' ""7 21' 1 1 K« .-. Si + S2+'S3+- + 'Sn=g22^"~' + 722-."'-i- 23; = J(2»»-l) + i(2«''-l)-g. 231. The probability reqtiired is the sum' of the last two terms in the (2 1\^ ^ + - J ; and therefore is equal to imh% 6 11 '"^243- 232. Subtract the second equation from the first ; thus 2z {x-y) — a^-l^; so that x-y = — 5 — . Substitute in the third equation; thus zi+^^LZp^=c\ or 4^-4A2+(a2-62)2=0; that is, (22= - cy = ( - a2 + 62 + c2) (a= - 6" + c") ; hence iz^=2c^±2 J[-a^ + b^ + c^]{a''-b'' + c^) = { ± J-a' + b^ + c^d^ Ja^-b^ + c^}\ 233. Let fc denote each of the given equal fractions; then x^-xy -xz = ak, y''-yz-yx = bk. Subtract the first of these equations from the second, and multiply the result by z; thus we have H{b-a)z = (x-y) [z^ -xz- yz) = (x-y)c'k. .: ex - cy + {a - b) z = 0. 515.] MISCELLANEOUS EXAMPLES. 337 Similarly bx + {a-c)y-hz = 0. Thus X -.y : z = a(b + c-a) : h(c + a-li) : c(a + b-c); by substituting for x, y, z in ax+by + cz = 0, we obtain a^ + b^+c^=a''{b + c) + b^(c + a) + c^{a + b). 234. If a is one root, tben - a is another root ; hence a''+pa^ + qa + r=0, and a^-pa^ + qa-r=0; from which equations we have a? + qa=Q, anipa' + r^O; thus (i^= -q, and therefore 2)2 =r. 235. (1) The scale of relation is (1 - x)*. [See Art. 398.] S = l + 8x + 21x^-+ 64xS + 125a;^ + ... -4xS= - 4a; - 32s2- 108x3 -25Ca;*-... ex^S= 6x3+ 48x3 + 162x« + ... -ix'>S= - 4x3- 32X''-... x^S= x* + .... On the left-hand side, it is easy to prove that the coefficient of x» = - 4m3 + 6 (re - 1)' - 4 (n - 2)' + (re - 3)3 = - (ra + 1)^ ; the coefficient of x-'+i = 6re3 - 4 (re - 1)' + (« - 2)3 = Sre^ + 6«2 - 4 ; the coefficient of x"+2 = - 4n3 + (re - 1)' = - 3ji3 - Sre^ + 3re - 1 ; the coefficient of x"+3_„3_ 5n2 + l2re + 8 (re3 + Gji^ + 12re + 8) - (re^ + re^) (2) We have re2 (re + 1)3 (re + 2)3 re2(re + l)3(re + 2)s (n + 2)3-re2{re + l) 1 1 .n^ (re + 1)3 (re + 2)3 re2(re + l)3 (re + l)2(re+2)3' thus «n=''n-''«+i. wheret)„= ^^ ^^^^3 ; and therefore S„ = rr^—~, - , , ,,,/ ■ om • " 1-^.23 (re+l)''(re + 2)3 236. In the identity (l + a3x*)(l + a«x3)(l + a9xi6)... = l + ^4X'' + ^8i' + /li2xi2+...+^4„x*'+.... Write a^x^ for x; then we get (1 + aW) (1 + o'x") (1 + ai7a;32) . . . = 1 + A^a^x^ + Jga^is + ,+ 4^^a2»x8n + ___ .-. 1 + ^4X4 + JgxS + . . . + J8„x8" + A 8„+4x8»+4 +' . . . = (l + a3x<)(l + ^4a2x8 + ^8a4xi6 + ...+44„a2»xSn+_ ). Equate coefficients of x^"; then Ag^=A^„a^''. Again, equate coefficients of x^""*^; then ^8?v+-4=-^4n''°"- o3=a348„. H.A. K. 22 338 MISCELLANEOUS EXAMPLES. [PAGE Now Ai=a.^; A^=a'A^—a^\ A-i^=a^A^ — ifi; A-^=a*A^ = a!>; A^ = a^A-^^=a>^; A,_^=aPA^=tt>'^; A^ = a?A^=aP; A^^=a!^A-^^=aP ; A^ = d?A^^=d^. Hence the first ten terms are 1 + a%* + a^a^ ■\-aH'^^+ o V^ + d>-V + d>H^ + dPx^ + dPai^ + a^^x^. 237. Let X and y miles be the distances from ^ to B and B to C; and suppose that the man rows u miles per hom:, and that the stream flows V miles per hour; then we have u u+v u u-v u+v while it remains to find , u-v From the above equations we have 11 V 1 — -— = s; thatis, 4ro = «2_t.3. u-v u+v 2 ' * ' and ^^ = 7-: that is, 4j;a;= «(« + «;) ; u u + v i \ I > hence by addition, iv(x+y) = (u + v){^u-v); i.v I 2m-ij x + y 11 ,, , „ therefore — -j — = ^ = -r- ; so that u = Cv. Thus a;+y^x + y M + ^<^11^^7^gl7 «-■!; u + u'm-j) 4 5 20' 238. Here, with the usual notation, we have J'„=2p„_i + 3;)„_2; thus the numerators of the successive convergeuta form a recurring series whose scale of relation is 1 - 2x - Si". Put Sp=pi+ p^x+ p^^ + ... then -2xSp= -ip^x-'ip^'^- ... -33^Sp== -Spjx^-... •• ^ l-2x-3a;2 ~ (l-3s) (1 + x) ' and therefore 516.] MISCELLANEOUS EXAMPLES. 339 In the same way -we may shew that '-=^^)-W^y and,„=l!3n+i-(-l)««}. 239. The equation cannot have a fractional root, for all the coefiBcients are integers, and that of .r" is 1 ; it cannot have an even root, for/ (0) or p,^ is odd, and hence / (2m) will be odd, since all the terms but the last are even. It cannot have an odd root ; for if z is odd, a"=an odd number = an even number +1. Hence /(s)=an even number + l+^i+^2+ ... +jp„ =an even number +/(!) = an odd number, and therefore cannot vanish. Thus the equation cannot have any commensurable root. [This solution is due to Professor Steggall.] 240. (1) By squaring we obtain ax + a + bx+p+2^f(ax + a) {bx+p)=cx + y; from this equation by transposing and squaring, i{ax+a)(bx+P) = {{c-a-b)x + {y-a-p)}^; this equation reduces to a simple equation, if 4a6={c-a-6)'; that is, if ±2,ya6=e-a-6; or c=a+6±2 ,^06; whence ^c=±,^/a ±,^6. (2) By transposition, j6jp-lax-7 + Jii^-8x-ll=(2x-3)+sj2x--ox + 6 (1). Xow we have identically (6x= - loj; - 7) - (4xS _ 8x - 11) = (^x - 3)s - (21= - 5i + 5) ; hence by division, j6j^-15x-1-,Ji3?-Sx-U = {2x-3)-J'h^-ox + 5 ... (2). From (1) and (2) by addition, J^-^-lox-l = 2x - 3 ; whence 2jS_3x-2=0; sothata;=2 or --. 241. At the first draw he may take 3 red, 3 green, or 2 red and 1 green, or 1 red and 2 green. In finding the chance that at the final draw the three balls are of different colours we may evidently leave out of consideration the first two of the above cases. 3x*C 9 The chance of each of the other cases = c^ ^ = -=r, ■ sCj 20 0-1 ^. 340 MISCELLANEOUS EXAMPLES. [PAGE Then after the 3 blue balls have been dropped into the bag, there are either 2 red, 1 green, 3 blue ; or 1 red, 2 green, 3 blue. ■I 9 q /» In each case the chance of drawing one of each colour = " ' =— . 9 6 27 .-. the chance of the required event = 2 x — x ^ = :j----- . Hence the odds against it are 73 to 27 ; thus he may lay 72 to 27 or 8 to 3 against it. 242. Here f{x)=x*-7x^ + 4x-3, and /' (a;) = 4a;S - 14a! + 4. Now Sj is equal to the coefficient of -; in the quotient of /' (x) by /(a;). 1 4 + 0-14+ 4 + 28-16+ 12 7 0+ 0+ 0+ 4 0+ 98- 56+ 42 3 0-84- 48... + 770... 4 + + 14-12 + 110 + 140 + . Hence Sj= 140. [Compare Art. 563.] 243. We have a9-^6'-i'cP-«=l. [V. a. Ex. 27.] Hence {q - r)log a + {r - p)logb + (p - q) logc = 0. Again, (q-r)lic + {r-p) ca + {p- q)ab = 0. [VI. a. Ex. 8.] By cross multiplication we see that q-r, r—p,p-q are proportional to a(b-c) log a, h{c- a) log 6, c{a-b) log c ; whence the result is evident, since the sum of q-r, r-p, p-q ia zero. 244. Denote the numbers hy x, y, z,u; then x-y + z + u=S; (1) a:2+3/2-z2-M2=36; (2) xy + zu=i2; (3) a^-y^~z^-u^=b (4) Prom (2) and (3) we have {x-y)'-{z + u)^= -'iS; dividing this equation by (1) , we have {x-y)-{z + u)= - 6 ; hence x-2/=l, and z + tt = 7. Now x^-y^ = {x-yY + Sxy {x-y) = l + 3xy ; and z^ + u^=(z + u)^-3zu(z + u) = 34:3-21zu; hence from (4), we have xy + lzu= 114. Combining this with (3), we find x!/ = 30, zu=^V2. Thus x-y = l, x!/ = 30; z + u=T, zu—Vi. 516.] ' MISCELLANEOUS EXAMPLES. 341 245. We have T^=aT^i-l>T^; ~ Jn=i '-^n ~ ^n+1 -^ n-l) =i;^{V-T„_i(ar„-6rH-i)} = r^(rn'-"^nl'n-l+6n-l) = 6^ (n-i - «r„_i T„_, + 6TV2) = Ti^ - aTir,, + bT^, which is independent of n. 246. (1) We have yz + zx + xy = -^ = — ; also (i+2/ + z)'' = 62 + 2(j/« + 2x+a;!/) = 6^4 . a Now x^ + y^ + z^ -Zxyz = (x +y + z) {x'^+y' + z'^-yz-zx — xy); or a' (c3 - 3d3)2 = {ab^ + 2d^) (ab^ -d^y-. 247. We shall shew that the roots of the equation X* -px^ + qx^ -rx + s=0 will be in proportion provided s=— . Let a, b, c, d be the roots, and let - = 3 = ft. a Now a + b + c + d=p; abc + abd + acd + bcd=r; abcd=s; or (b + d){l + k)=p; ftdfc (6 + rf) (1 + J)=r; b''d^h'=s. T T^ Whence - = bdk=Js; that is, -5=s. p ^ p^ In the case of the equation X* - 12x3 + 47a.2 - 72x + 36 = 0, we have (i + d) (l + /c) = 12; h(b + df + b8 + Ji%SJ = 16mV(1-u8) (1-2,8). 252. (1) Here x, y, z are the roots of t^-Zpfi + Zqt-r=0. Let u=y+z-x; then u=(2/+« + a;)-2a;; so that we may put u = Sp-2t, or 2f=3p-«; hence (3p - m)^ - 6p (3j3 - m)2 + 12 j (3^) - «) - 8r = 0, or V? - 3jjm2 _ (9^,2 - 12g) tt + 27^)3 - 36^)2 + 8r = 0. The product of the roots is - 27j)' + 36pg - 8r ; which proves the first part of the question. (2) For the second part we have to find the sum of the cubes of the roots of the equation in u. Denote the roots by Uj, Mj , M3 ; then 2« = 3^; 2m2= (s«)2 - 22«iUa= (Zpf - 2 ( - V + 12?) = 27^^ - 2'kq. Again by writing % , u^, u^ successively for u and adding, we have SM3-3^2™2_(9p2_i2g)2u + 3(27y-36pg + 8r;=0; .-. 2uS = (81i)3 - 72p}) + (27p* - 36p2) - (81^^ _ loSjjj + 24r) = 21 f - 2ir. 346 MISCELLANEOUS EXAMPLES. [PAGE 253. The coefficient of !C* = o^ (6 + c)2 - iw'bc =a'(b-c)K The coefficient of yV = 26c (a + 6) (a + c) - iabc (6 + c) = 26c (a - 6) (a-c). Let o(6-c) = 4^ 6(c-a)=JS2, c{a-h):=C\ then the given expression = A^x* + BY + C*^* - 2B''C2!/2«2 - W^AHH"^ - 2A^ffx^y^ = - {Ax + By + Cz){- Ax + By + Cz){Ax - By + Cz) {Ax+By - Cz). 254. If X, y, z are not integers, we can find an integer p which will make px, py, pz integral. The expression x''"'y"zJ"' is the product of px+py+pz factors, and the arithmetic mean of these factors is px' +py^ +pz^ x^ + y^ + z^ px+py+pz x + y + z ^ — j > xP^yPi!^'. [Art. 253.] By taldng the p*^ root we get the required result. For the second part see solution of Ex. 6. XIX. b. 255. The expansion of (l-4j^)~ 2 ia 1 +i'i2/ H-i'sV" +i'3!/' + - +2'r2/'" + ■ • • . |2r where Pr=j-i^. [See Example 33. XIV. b.] If we put for y its equivalent x{l + x)~^, we shall have a series whose general term is In this and all subsequent terms pick out the coefficients of ic" and equate their sum to the coefficient of a;" in (1 +x) (1 - x)-^. Then 2 = T^(-l)-.?r^(?yiMH:X^. • !-?( ir-- ^- /(2r + l)...(ro + r-l) r-l^ lit' = S (-1)"-'- m-r r-i Ir |r- 1 \n-r 518.] MISCELLANEOUS EXAMPLES. 347 256. (1) Substitute z= ~{ax + by) in the second and third equations; thus x{ax + by) = ay + b, and y{a!ii + by) = bx + a. OiX _ Ji Oi 3! ~~ & From the first of these equations, y= _ ; so that ax + by— r— • By substituting in the second equation, we have (ax^ - b) (a'x - b^) = (bx + a){bx- a)^ ; whence (a'-6')(a;'-l) = 0; and therefore the values of x are 1, w, by'. ax^ — b The values of y and z are obtained from y= — —j— ; z= - (ax + by). (2) From the second and fourth equations, we have (x+yf-(z-uY=96 (1); but (x + y) + (z-u) = 12; and therefore (a; + 2/) - (z-m) =8; whence x + y = 10, and z-u=2. Now x^ + y^={x + y)^-3xy{x + y) = 1000-30xy; and z^ - «^ = («-«)' + Szu (z-u) = 8 + Gzu. By substitution of these values in the third equation, we obtain 992-30x2/ -62tt=218, or Sxy+zu=129. From this equation and the fourth of the given equations, we find xy=21, and ZM = 24. The solutions are therefore given by x + y = 10,) . z-u= 2,) xy = 21-S zu=2i.\ 257. Put p = q+x, where x is very small ; then (n + l)p + {n~l) q 2nq + {n + l)x {n-l)p + (n + l)q~ 2nq + (n-Vjx Taking in terms as far as a?, the left side of the given equation / n + 1 \( n-1 (ra-1)^ 2 (?i-l)3 _, ) and the right side of the given equation nq 2n^q^ 6n*j" 348 MISCELLANEOUS EXAMPLES. [PAGE In these expressions the difference between the coefficients of x^ Jn-l)^-(n + l){n-l) + 2{n-l) ^^ The difference between the coefficients of x' is _ 6fa+l)(w-l)^-3(n-l)3-8(n-l)(m-2) _ (w-l)(3wg-2K + 7) Thus the difference is of the order -^ , and as - is a, decimal beginning with r - 1 ciphers, -3 wiU be a decimal beginning with at least 3r - 3 ciphers. 258. Denote the prices of a lb. of tea and a lb. of cofiee by r and y shillings respectively, and the amounts bought by u and v lbs. respectively; then the amount spent =ux + vy shiUings. 5 4 9 Hence -ux+ -vy = jzr{ux + vy); D ±1. ,, . . UX Vll U V that is, ■;;- = ^ , or ;7- = TT- . 6 5 by 5x Again vx + uy=ux + vy + 5; so that (a; - ^) (?; - a) = 5. Also M-i-« = 54, and 6y~2x=5. Hence v + u ^ 5i(x-y) ^ 27 (x-y) _ v-u 5 Sy-x ' but v + u 5x + Gy v-u 5x — Gy' therefore 5x+_% ^ 27 (x - y) ox -by oy — x or 70x2-153x2/ + 722/2=0; whence (2a;-32/){35x-242/) = 0. Combining 2x-Sy = and 6y-2x = 5, we have x = 2J , 2/ = 1|- By hypothesis tea costs more than coffee, and therefore 3!ix-2iy=0 is inadmissible. 259. Here 2s^={l + 2 + 3+ ...+ny-(V + 2^ + S''+ ...+n^ ~ 4 6 ' ••• s„ = 23(«-l)«(™+l)(3n + 2); _ (to-2) (ro-l)TO(3ra-l) •■• s»-i- 24 ■ 518.] MISCELLANEOUS EXAMPLES. "349 [n 24 |»-3 2i \\n-i^ \n-3j 260. If T is ^^^ value of each ratio, we have pa' + 2qab + rb^=lcP (1), pac + q(bc- oF) - rob = kQ (2), pc^-2qac+ra'' = kR (3). MaHiply (1) by a, and (2) by 6 ; then by addition, we have pa{a^ + bc) + qb{a^ + bc)=]c{aP + bQ); that is, (pa + qb){a' + bc)=^k{aP+bQ). Similarly from (2) and (3), we obtain (pc - qa) (a? + bc) = k{aQ, + bR) ; pa^ qb _ aP + bQ , _ pc — qa aQ + bR ' p Pa^ + 2Qab + Rb' *^^*'^' q~Pac + Q(bc-a')-Rdb' If we eliminate p instead of r, we find r_ Pc^-2Qac + Ra' q~ Pac + Q {bc-d')-Rab ' 261. Let a, jS, 7 denote the roots of the cubic equation a;' + gx + r = 0. Multiply this equation by x", substitute in succession a, jS, 7 for x, and add; then (a"-W + /S»+5 + 7"+') + q (a"+i + j3"+i + 7"+') +r{o" + /3»+7«)=0; but 5=j37 + 7a + a^=|{(a + i3+7)=-(a= + /S= + 7=)}=-2(o= + /32 + r°); and r = - a^y ; whence the result at once follows. 262. The expression Z: (o - /3)^ (7 - 5)^ consists of three separate kinds of terms, and when multiplied out and arranged is easily seen to be 2i;a2j32 - 22aj37'- + 12Za,87S. 350 MISCELLANEOUS EXAMPLES. [PAGE Now Sa2/32=(Sa^)2-22a. So^7 + 2Saj375; and Sa^v^ = So . Sa/37 - 4Sa;375. Thus the given function becomes 2 (Sa^)2 - 62a . Sa|37 + 24S 0^78, or 2q''-6pr + 2is. [This solution is due to Professor Steggall.] 263. Denote the number of turkeys, geese, and ducks by x, y, z re- spectively ; then we have a;2 + 2y2 + j.2=211, and x+y+z=23. Eliminating z, we obtain x'' + xy + y'-23{x + y) + 159 = 0; hence 2x=-{y- 23) ± J -Sy^ + i6y -107. Thus -Zy'+iey-107=u' say ; that is, 3y''-i6y + 107+u'=0 (1) ; whence 3i/ = 23±^/208-3m2. Thus 208-3u^=f2; hence u must be less than 9; by trial we find that u=2, 6, 8. On substituting in (1), we have 3!/2-46j/ + 107 = -4, or -36, or -64. The integral values of y found from these equations are 3, 11, 9. Ill 111 264. If a? + b^ + c^=0, then o + 6 + c = 3a%V ; hence from the given equation, we have 1 S {{y + z-6x) (z+x-6y) (x + y -8z)Y=-G {x + y + z); and therefore (y+z-8x)(z + x-8y){x+y-8z) = -8{x + y + zy. Puta!-l-2/-l-z=jp, then we have {p-dx)(p-9y) {p-9z) = -8p^;, that is, p^ -9p^(x + y + z) + 81p (yz + zx + xy) - 729xyz = - 8^^ ; or jfi - 9^' + 81p (yz + zx + xy) - 'J29xyz = - 8p^ ; or {x + y + z) {yz + zx + xy)~9xyz=0; that is, x^ {y + z)+y^ (z + x) + z^ (a; +2/) - 6xyz=0; whence the result at once follows. 265. We have — — = — — = -r ; x + a x + c x + a x + b {a-c)x _ [d-b) X {x + a) (x + c)"" {x + b) (x + d) ,(1). 519.] MISCELLANEOUS EXAMPLES. 351 Thus (a + b-c-d)x'+2{ab-cd)x + ab{c + d)-cd{a + b) = (2); or a;=0. If the given equation has two equal roots, then either equation (2) has two equal roots, or it has a root equal to zero. In this latter case, the absolute term vanishes, so that ab(c + d)-edla + b) = 0, or - + -=- + -. ^ ' \ I • abed The remaining root is then — -, , which is equal to , ® a+b-c-d ^ a+b' „ db cd ab — cd for a+b c+d a+b~c-d If equation (2) has a pair of equal roots, then {ab-cdf={ab(c + d)-cd{a + b)} {a + b-c-d) (3); thatis, {a-c){a-d){b-c){b-d) = 0; for equation (3) is satisfied when a = c, a=d, b=c, b = d, and is of two dimensions in a, and also of two dimensions in b. Thus one of the quantities a or 6 is equal to one of the quantities c or d. Suppose that a=c; then each of the equal roots = r -,= - a. a+b-c—d Similarly in the other cases. 266. (1) By multiplying together the second and third equations, we have yz+zx + xy = a%; thus x + y + z=ab, yz+zx + xy = a^b, xyz=a^; hence x, y, z are the roots of t^-abf^+a%t-a^ = Q; or (t-a){f + at + a^-abt] = 0. (2) From the first and second equations, we have z (ay -bx) = ax -by ; hence by substituting io. ax + {bx + c)z=a + b + e, -we have (bx + c) (ax -by) , , , ax+^ '-^ ^=a + b + c; ay - bx thatis, {a''-b')xy + x(ae + ab + b^ + bc)-y(bc + a^ + ab + ac)=0; or (a-b)xy + x(b + c)-y(a + c)-0; {b + c)x whence y = -, — — j — ,- — ;->- - (a + c)-(a-b)x 352 MISCELLANEOUS EXAMPLES. [PAGE Substituting in the equation cxy + ax + by=a + b + e, we liave (b + c) (cx + b) X , — .-,1 1\- =a + b + c-ax; (a + c)-{a-b)x or {be + c''-a^ + ab)x' + Px-(a + c){a+b + c) = 0. Now the given equations are obviously satisfied by x = l, y = l, z = l; , J.1, 4,, , J (a + c)ia+b + c) a + b + c hence the other value of x=^, y — ; — r- = ; . a^ — c^ — ab — bc a — b — c 267. Let f{x) = {x-a){x-l3]...{x-e), [x) be an expression of degi isolved into partial fraetior 4>(^) 0(") and let (a;) be an expression of degree not above the fourth ; then {x) -hf {x) may be resolved into partial fractions. We have, as usual, T7-^ - 7 w 5T7 w fw ^ + similar terms. f(x) {x-a)(a-/3)(a-7)(a-S)(a-£) If x=0, we obtain 0(0) ^ 0W , tM . /(O) _a(a-^)(o-7)(a-S)(a-e)'^-|3{j3-a)(/3-7){/3-5)(^-e)"^- Por the given example, we take ip{x]=x\ so that (0) = 0; a3 ^ "^ (a-/3)(a-7)(a-5)(a-e) + (3-a)(p-7)(|3-5)(;3-6) + --"- The more general theorem, which can be proved in the same way, is found by resolving x^{x)-i-f{x} into partial fractions; where f(x) is of n dimensions in x, and ip{x} oin-2 dimensions. In this case 0(°) , 0W I ^0 (a-/3)(a-7) "^ (^ - a) (;3 - 7) "^ [This solution is due to Professor Steggall.] 268. Let X, y, 2 denote the number of Clergymen, Doctors, and Lawyers respectively ; u, v,w their average ages ; then ux+vy + 'Wz = 'iWO; ux + vy + wz „„ ^, ^ ■ =36; BO that a; + « + z=60. x+y+z " ux + vy=39{x + y); vy + wz = S2^{y+z); ux + wz = Z6^{x + z). From these three equations, we have 2 (ux + vy +wz) = 75§ a; + 71/^ y + 69^ z. But ux + vy + wz = S& (x + y + z); therefore 72;!; + 72j/ + 72s = 75f a; + Tlj^ «/ + 69 Jf 2 ; or 121a! -9?/ -862 = 0. 619.] MISCELLANEOUS EXAMPLES. 353 The increased average age is ; ° * x+y+z but this is equal to 5 ; hence ix-y-2z=0. From the last two equations, we have by cross multiplication, t = ^ = H i but a; + y + z = 60; hence x = 16, y = 'ii, 2=20. Again 16u + 2iv = 39 x 40 ; that is, 2u + 3» = 195 ; 24d + 20w) = -— x44; that is, 6u + 5«) = 360; 16m + 20w = -5-x36; that is, 4m + 5u = 330; whence m=45, i;=35, w = SO. 269. Let the two expressions be ax + by and cx + dy; then we have the identity a„a;* + ia-^x^y + ... + aty''= {ax + by)* + {cx + dy Y; hence, equating coefficients, a„=a*+c*, aT^=a^bt-cH, a^=a%'' + cH'', as = ab' + cd', ai=b* + dK Prom these equations, we obtain Oodj - Oj^ = a^c' (ad - 6c)'' ; aiOj - a^' = abed {ad -bc)^; a^^-a^=bH^{ad-bcf\ and therefore the condition required is (a„a2 - fli^) {a^a^ - aj") = {a^a^ - a^")". We may also proceed as follows : bda^ +aca^=bd {a* + (!>■)+ ac {a%^ + cH^) = {ad + bc) {a^b + cH) ; that is, bdaa-{ad+bc)ai + aca2=0. Similarly, bda^ -{ad+ be) a^ + aca^ = 0, and idnj - (ad + be) Oj + aca4 = ; from which, by eliminating bd, ad + bc, ac, we have a^ flj a2 =0. Oj Oj Oj ^2 H "-i A general theorem, of which the above is a particular case, is proved in Salmon's Higher Algebra, Arts. 168, 171. [This solution is due to Professor Steggall.] H. A. K. 23 354 MISCELLANEOUS EXAMPLES. [PAGE 270. We have (y^ + w' + w''){z^+u'' + v^) = b'^c^=[vw + uy + uzy; on reduction -we obtain (m^ - yz)' + {wu -vy)^ + {uv - wz)^=0. Since the roots are real we must have w'-yz=0, wu-vy = 0, uv-wz = 0. From the other equations we obtain similar results ; hence v?=yz, v^=zx, w'^ = xy; vw = ux, wu=vy, uv=wz. Substituting these values in the given equations, we have x{x + y + z) = a'', u{x + y + z) = ic, y{x + y + z) = b\ v {x + y+z) = ca, 2 (x + 2/ + 2) = c^, w(x + y + z)=ai. 'H.enae {x+y+zY=a' + i' + c'; and thus a^ ic ■T= ± - ; u= ± — J a- + b- + c^ iJuF+W+c^ 271. We have m + 3 letters in all, of which three are the vowels a, e, 0. The consonants can stand in any of the m + 3 places so long as their position does not involve an inehgible arrangement of vowels. The vowels in any word can occur in the following orders : (1) aeo; (2) oea; (3) aoe; (4) eoa; (5) eao; (6) oae. Now (1) and (2) cannot stand at all unless all the vowels come together. Therefore there will be 2 [m + 1 words which have the vowels arranged in this order. Now consider any one of the four remaining cases, such as aoe. Here oe must come together in any word, and a must precede oe. Therefore in considering the number of words possible with this arrangement, we have only to select two places out of Ji + 2, and then fill up the remaining n places with consonants. This gives rise to "^^Cj x In words. It will be found that each of the three remaining cases gives this same number of words. Thus on the whole the number of words is 2 |m + l +2(re + 2) (ji+1) [ra, which easily reduces to the required form. 272. We have a;2- 2^=22 _ ^2. that is, (a; + 2) (a; -2) = (2 + 1/) (2 -2/). This equation is satisfied if k(x + z) = l{z + y), and l{x-z) = k(z-y); that is, kx-ly + (k-l)z = 0, and Ix + ky -(k + l)z = 0. By cross multiplication, we obtain X _ y _ 2 _r 2« + J3~i2 ~ V + 2lk-P ~ k^ + V ~ 2 °^^' 520.] MISCELLANEOUS EXAMPLES. 355 273. Here the n"" convergent is -^ J ; hence u„=(2»-3)«„_i + 2(n-2)u„_j; that is, K„-2(n-l)«^i=-K_i-2(n-2)M„_^!; Uj - 2 . 2u2= - («2 - 2mi) ; whence, by multipUcation we obtain «„-2(n-l)«„_i=(-l)"-2(„2-2Mi). But j)i=l, j)2=l; 3i=l, S2=2; hence P»-2(™-l)i'„-i=(-l)»-i, S„-2(n-l)g„_i=0. Thus g„=2(n-l)j„_j=2»(«-l)(»-2)j„.2=... = 2"-i|«-l. Asaia y» 2p„_, (-1)- | m-l |«-2 |7i-l ' 2f„-i 2X-2_ 2(-l)"-'' |7t-2 ln-3 |»-1 [1 ^1 li ' hence by addition, -p^ - 2»-i= - 2«-2+ -j^ — - ... ; In — i. J o *^^t-' 2S=I^=l-i + 2%-2% + -= and therefore | = .-i. 274. (1) We have ^^^:i^= - „-^ + ^2 • Thus the series=a2 (-2 + 3)+^(~3 + i) + ^(~4 + 5) + — fx^ a^ X* \ 2/x^ X* x<' \ = {x+log(l-a:)}+-j-a:-y-log(l-x)|. 23—2 356 MISCELLANEOUS EXAMPLES. [PAGE (2) We have 12 (a + l)(a + 2)...(a + m) 1 f |» !" + ! 1. ~a~l\{a + l){a + 2)...(a + n-l) (a + l)(a + 2) ... (a + m)J ' 1 f |"+1 1 a-1 [ {a + l){a + 2)...{a + nj) 275. (1) Put 2x=u, 3y = v, iz=w; then uvw=-Z&, (m-1)(d + 1)(m)-1)=-12, (« + 1)(«-1)(m; + 1)=-80. Thus from the second equation uvw + (-vw + wu-uv) + (-u+v-w) - 1= -12; or (-vv}+wu~uv) + (-u + v-w) = '2^. Similarly from the third equation (vw -wu + uv) + (-u+v~w)= -43. From the last two equations, we have vw-wu + uv= - 33, . and -u + v-w= -10 ; that is, v(-w) + (-w)(-u) + (-u)v=Zi, and (-«)+»+(-!«)= -10. Thus -u, +v, -w are the roots of the equation «3 + 10*2 + 33* + 36 = 0, or (« + 3)(S + 3)(« + 4) = 0; and therefore -u, v, -w are the permutations of the quantities -3,-3,-4; that is, 2a, - 3j/, iz are the permutations of the quantities 3, 3, 4. (2) From the equations Sux - ivy = 14, vx + uy = 14, we have (3m2 + 21)2) X = 14 („ + 2v), and {Sw' + 2v^} i/ = 14 (3u - v). But 3«2 + 2i;2 = 14; .•. x = u + 2v, y = 3u-v; :. (M + 2t!)(3M-'!)) = 10OT; that is, 3M2-5Mti-2))2=0, or (u-2v)(Zu + v) = 0. Taking u=2v, and combining with 3u^ + 2v^=li, we have M=±2, «=±1. Similarly from «= - 3m, we have "=-jx/§' "=v^ 521.J MISCELLANEOUS EXAMPLES. 357 276. Keeping the first row unaltered, multiply the second, third and fourth rows by a ; this is equivalent to multiplying the determinant by a?. Next multiply the first row of the new determinant by 6, c, d and subtract from the new second, third, and fourth rows respectively : thus o3A= z=a?\^ Thus the remaining factor is the last determinant, which reduces to a^ + h'^ + c^ + d' + X. a2 + \ ab ac ad -b\ a\ -c\ a\ -d\ aX a^ + \ b c d -b 1 -c 1 -d 1 277. Here and therefore Now Thus Also that is 2a=— pi, 2,ab=p2, ^abc= —p^; 2a2=^^2_2p2. (2a)'= Sa' + 32a26 + 62a6c. -p3=Xa?+S'2a'b - &Pi. Sa3.Sa = 2a3 + 2o26; -Pi(i'i'-2i'2) = Sa3 + Sa26; by eliminating Za?b from the last two equations, we have 22o3 + 6p3 = - 3pi {pj^ - 2p^) +p^^ i or 2a^= -p^^ + Spjp^-Sps- [Art. 522.] The equation whose roots are - , ^ > - > ^ a b c 111 .-. - + -r + - + ...= - a c = 0; Pn ' thatis, -Pi + ^'^=-"'~{Pi-^P-:^- .p^ + ^-'-Pj^ip^^ " b Pn 358 MISCELLANEOUS EXAMPLES. [PAGE 278. Separate . ^ into its partial fractions ; thus T 5 = 1 +T-. r^= 1-a; -i + 9!(l + ii: + a;=)-l 1-x'l-xl + x + x-^ ' ^ ' = {l + x + a;2+a;' + ...} X a? x^ "*" l + x~ (I+^"*'(TTxp~ ' '■ Again ^^^ = (l + ix)(l + x^ + x^ + x^+ ...) (2). In this last expansion every term is of the form x'" or a:^"+'. If we expand each term of (1) we shall have l + x + x^+...+x'^+... +x{l~x + x^-... + (-lYx''+,..) ~ x^ {\-2x + ^x^ - ... + (-Vf (r + 1) x^ + ...} +x^ ji-3x+6»^- ... +(-ir ir±AHi±^) ^r+ ... j Now equate coefficients of x'"+2 in this expansion and in (2) ; thus = 1 + ( - 1)3"+! - ( - l)3''-i 3n + (- l)'»'-3 (^"-^) (^ri-1) on transposing the first term and dividing every term by (- l)3™+i vve get the required result. Or thus : By the Binomial Theorem, we see that (3m-2)(3w-l) (3w-4 )(3ra-3)(3m-2) 1. rf», j^;^ . j^^2^-3 , are the coefficients of a;'"+i, a?'^-^, a:^''-^ , ... in the expansions {l-a;)~S (1 - x]~^, (1 - a;)"*, ... , respectively. Hence the sum required is equal to the coefficient of x^^+i in the expansion of the series 1 x^ X* >. + -, 1-x (l-a;)" {l-xy and although the given expression consists only of a finite number of terms, this series may be considered to extend to infinity. 521.] MISCELLANEOUS EXAMPLES. 359 But this last expression is a G.P. ■whose sum 1-x \ 1-xJ 1-x + x^ 1 + x^ = (l + x)(l-a;3 + a;8-x» + ... + (-l)''a;'" +...). Thus the given series = ( - 1)". 279. Let X, y denote the number of shots fired by A and B respectively; and suppose that A killed 1 bird in m shots, and £ killed 1 bird in v shots ; then - and - denote the numbers of birds killed. u V Hence we have the following equations : a:" + 2/2 =2880; xii= ; that is, «■!; = 48 ; •' uv - + ^=10; u V V u From these last two equations, we have a;2 + s" u - = lOic - 5y ; and therefore u{2x-y) = S16; x" 4-y^ ^ = lOy + 5x ; and therefore c (2j/ + x) = 576. .-. uv {2x -y) {2y + x) = ol6 X 576 ; .-. (2a;-2/)(2y + x) = 12x576. . (2x-y)(2y + x) ^ 12x576 _12_ aP+y^ ~ 2880 ~ 5 ' that is, 2x2 _ i^^y + 22i/2 = ; or (x-2y){2x-ny)=0. The equation 2x - llj/ = does not lead to integral values of xandy. Putting x = 2y, we have a! =48, y=24. 48 24 Thus — H = 10, and Mu = 48; whence u=8,v = 6. u V 280. By Art. 253, we know that o3 + 63 + c3>3a6c; .-. 2{a^ + l? + c^)^>18a'hV (1); and 3(a' + 63+ca)2>9a6c(a3+63 + c3) (2). 3G0 MISCELLANEOUS EXAMPLES. [PAGE Also (63 _ c3)2 + (c3 -a^f+ (a? -bY>0; that is, a^ + W + c^> ft'c^ + c^'a? + a%'^; whence (a? + b^ + c^f>Z (b^c^ + c^a? + a?b^) ; and therefore i(d^ + b^ + e^f>^(bV + c'a^ + a?b^) (3). From (1), (2), (3) by addition, we have 8(a3 + 63 + c3)2>9{2a262c2 + a6c(a3 + i3 + c3) + 6V + c3a3 + a363j ; that is, 8(a3 + 63 + c3)2>9 (a? + hc) if^ + ca) ^c^ + aft). [See Solution to XXXIV. b. 28.] 281. We have u„=(m + 2)«„_i-2™„_2. [Art. 444.] Therefore m„ - 2m„_i = « («»-i - 2u„_2) . Similarly, «„_i - 2m„_2= (re - 1) (m„_2 - 2«„_3), «3-2it2=3(«2-2Mi); Now 1)1 = 2, 2i = 3; i)2=8, (?3 = 8; hence y„ - 2p„_i = 2 [re, j„ - 2?„_i = \n ; 2p„-i-2V»-2 = 2-^ ti. 2j„_i-222„_2 = 2 |b-1; 2"-2iJ2 - 2"-!^! = 2»-i [2, 2"-2 g,^ _ 2»-i3i = 2"-2 [2 ; 2"" Vi = 2" [1, 2"-i 2i = 2"-i . 3 = 2»-i 1 1 + 2" ; whence by addition, ;)„ = 2" []^+ 2"-! |2^+ 2'^2 1 3 + . . . + 2 |n ; 2„=2» + 2"-i ]l + 2"-^ |2_+ ... + \n. and therefore i'n= 2g„ - 2"+''; V 2"+i .-. ■'-i= 2 ; and g„= S" 2'" [ rc-r. 2" 2 + 2^^ 23+ liJ+"- If v„ denote the n* term of this series ln-1 , Ira JJ„_i=J= and v_ = L. 2'»-i 2"' hence —2- = - , and the series is obviously divergent. 2«+i n Thus Zii'm. = 0; and therefore iim.-£-i = 2. in Sn 521 ] MISOELLANKOUS EXAMPLES. 361 282. We have ?32+3 _ J_ J^ Ji_ & , The first three convergenta are 1 6 bc + l a' a6 + l' abc + a + c' . Psn+3^ (''e + l)g3n + fty8n and since the oonvergents are in their lowest terms, i'3n+3 = *2'3»+(6. y 9 Q8 7 6 5 4 3 2 ■■v / '\ ^^ X ^^ ,/ \ ^\ / ^\ ^^ /■ \ /' \ \^ \ ■-, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 A P B Then any point in the figure represents a combination of two of the numbers 1, 2, 3, ...n. But for our purpose we must exclude the points in 521.J MISCELLANEOUS EXAMPLES. 363 the diagonal AD wliioh represents repetitions, (1, 1), (2, 2),.... Now a combination of two points {xy), {x'y') will be suitable if x + y=x'+y'; that is, we must select points from the same cross diagonal, such as PQ in the figure, since all such lines are represented by an equation of the form x + 2/ = constant. But in any cross diagonal each admissible combination occurs twice over; in PQ, for example, we have (1, 8) (8, 1); (2, 7) {7, 2)...; hence in any cross diagonal points must be chosen from one half of the line only. (1) Let n=2m. Begin with the central diagonal and proceed towards the point A. This diagonal has m available points, and therefore from these we have — ^-^ — - combinations. Each of the next two diagonals con- tain m - 1 available points, and from each diagonal we get ^ ^-^ combinations, and so on. The diagonals (2, 2) and (3, 3) give no combi- 2,1 nations, and therefore the last term of the series is — ^ , and this term like the rest occurs twice. Therefore on the whole, remembering that the same combinations also occur above the central diagonal as we proceed towards D, the number we shall have m(m-l) f(m-l)(m-2) (m-2)(m-3) 2.11 = 2 +*] 2 + 2 + - + "2"1" m (m — 1) 2 i + 2{1.2 + 2.8 + ... + (m-2)(m-l)} m(m-l) 2, „. , ,, 1 , ,,,, _. = — ^-^ — - + 5 (m - 2) (m - 1) m=- ns (m - 1) (4m - 5). (2) Let ;i=2m + l. Then the diagonals beginning with CB and coming down towards A contain m, m, m-1, jre-1, ...2, 2 available points respectively. Thus the combinations arising from these diagonals are respectively m(m-l) m(m-l) (m-l)(m-2) (m-l)(m-2) 2.1 2.1 2 ' 2 ' 5 ' 2 ' ■■■ ~2~' "2~ ' Also as before, each series of points except the central one occurs again as we pass from CB to D. Thus the whole number of combinations _m(m-l) I m(m-l) (m-l)(m-2) 2^) 2 ( 2 "^ ■ 2 -t-...-t-^. jj j = ^(m-l)m(4m + l) on reduction. 364 MISCELLANEOUS EXAMPLES. [PAGE 284. From Ex. 21 of XXX. b. we have and «30(„)=J(l-l)(^l-l)(l-iy..+J(l-a)(l-6)(l-c)...; .■.6nv,,t>{n)-iv,'P{n) = n^(^l-l^(l-fj(^l-fj...=nMn); .'. v? - 6m«2 + 4^3 = 0. 285. Put «, 6. cforj/-2,s-x,x-2/ respectively ; then a + b + c=0, and we have identically (1 - at) (1 -- bt){l-ct) = l- qt^ - rfi, where q— -{bc + ca + ab) , and r = abc. Taking logarithms and equating the coefficients of t" we have -(a'' + 6" + c") = the coefficient of t" in the series n ' {qt^ + rt^) + ^ {qt' + rt'f + | (qf^ + n'f + ... =the coefficient of t" in f(q + rt) + - {q + rt)'^ + 5- {q + rt)^ + .... If n=6m±l, the only terms on the right which need be conyiiered are By expanding the hinomdals in this expression, it is easily seen that the coefficient of every term which contains t^i"-^ is divisible by qr, and the coefficient of every term which contains t6'"+i is divisible by q''r. Now since a + 6 + c = 0we have a^ + 6^ + c^= -2{ab + bc + ca); that is, (y-zf+[z-xf + {x-yy'=-2{ab + bc+ca), or x^ + y' + z'-yz-zx-xy=- (ab + bc + ca) = q. Therefore (a - ?/)" + (y - 2)" + (z - a;)" is divisible hy x^ + y^ + z^-yz-zx-xy, when n is of the form 6m- 1, and by {x'^ + y^ + z^-yz-zx-xy)' when n is of the form 6ni + l. [Note. It is easily seen that several of the examples on pages 442, 443 are particular oases of this general result.] 523.] MISCELLANEOUS EXAMPLES. 365 286. For the sake of convenience let ua denote the quantities by u, b,c, d, e, ...; then as in Art. 253, we have mabcdef to m factors < a™ + ft*" + c^ + d^H- «'"+/'"+... to m terms macdefg < a™ + c™ + d™ + c™ +/™ + g™ + mabcfgh < a'" + fc" + c™+/" + 0" + /i"+ Ira-l By addition, mP < -. — ' — -S; for the number of times that each n - m m - 1 of the terms o™, 6"", c™, ... will appear in the sum is equal to the number of combinations of n — 1 things taken m — 1 at a time. 287. By eliminating x^, we obtain that is, q(x^ + q)= - Srx ; but x{x^ + q)=r; .: ?=-3j;, or 3x' + q = 0; X and this is the condition that the first equation should have a pair of equal roots. If each of the equal roots is a, the third root must be - 2a; hence q= - Sa^ and r^ -2a^. Thus the second equation becomes a;' + 9ax2 + ISa^x -25a?= 0, or (x-a)(x^ + Wax + 25a') = a; and its roots are a, - 5a, - 5a. 288. li p + q + r=0, then we know that p*+ q* + r*- 2q^r'' - 2ry - 2pV=0- Hence from the given equation, we have X* (2a2 - 3xY + ... - 2y'-z^ {2a^ - 3y^) {2a^ - Sz^) + ... =0; or arranging in powers of a, Aa^ (x* + y* + z*- lyH^ - 2z''x^ - ixh/"^) - 12a- (a;^ + y^ + z^- y^z^ -y'^z*- z'^x'' - z^x* - x*y^ - x'y*) + 9{x^ + y' + z^-2y*z*-2z*x*-2xY) = (1). 366 MISCELLANEOUS EXAMPLES. [PAGE Denote x*+y* + i:*-2yh^-2z^x^-2x^y^ by P; then 2a;6 _ :2xY =(x^ + y^ + z^) P + Gx^y^z' = a'P + Gx^y^z^ ; and Sa;8 - i'Zyh* = (»« + y^+zi + iy^z^ + izV + Ix'y'^) P + SxY^^ (x^ + y^ + z^) = a*P+8a'xYz^. Thua (1) becomes = 4a«P - 12a.^ {a'P + 6xY^^) + 9 {a*P + 8aVyV) -a*P. Thus P=0 ; but P=-{x+y + z) {-x-i-y + z){x--y + z) (x + y-z). 289. The equation _fi_,_f2_ , I gn ^1 {9-ai){9-a^)...{e-a„) e-6i e-6 ■^■••■^fl-6„ {e-b{,{e-b^)...{e-bj' ■when cleared of fractions is of the (m- 1)"" degree in B; and in virtue of the given equations it is satisfied by the n values a^, Ujt "3 ■•• "^ni hence it must be an identity. [Art. 310.] Multiply each side by 9 - ft; , and then put fl = 61 ; thua ^ (6i-ai)(&i-a2)...(6i-a„) ' (61-62) (61-63) ...(61- y This example ia an extension of Art. 586. 290. As in the example of Art. 498, the determinant of the left-hand side is the square of the determinant X y z y z X z X y for its constituents are the minora of the several constituents of this deter- minant. But the square of this determinant, formed according to the method explained in Ait. 498, is the determinant on the right-hand side. 291. Suppose that A, B, C could do in one day fractions of the work represented by u, 0, w respectively, and that they worked for x, y, z days respectively. Then we have the following equations : ux + vy + wz = l; 4:0{u + v) = l; 2vy + 2wz = l; {u+v + 'w)y = l; 2 = ux + iwz = l; x-y : a;-z = 3 : 5. 523.] MISCELLANEOUS EXAMPLES. 367 From the first three equations, we have ■ux-vy-wz = 0, UX + 3vy - ^z = ; ux vy wz whence — = -^ = — . O J x Subtracting the fifth equation from the first, we have u{x-y)+w(z-y)=0; , u w ^"* 31 = ^: hence 3z {x — y) + x{z—y) = 0, or 3yz — 4zx + xy = 0. Again 5(x—y) = 3{x — z), or 2x = 5y — Zz; also Syz=x(iz-y); hence Gyz = {Sy — 3z) {iz-y); that is, 5y^-nyz + 12z^ = 0, or {y-z){5y-12z) = 0. The root y-z must be rejected because of the equation S{x-y) = 3{x-z). Hence 5y = 12z, and therefore 2a; = 9z. Thus also so that 15u=12u = 10w. Substituting in (a + ») 40 = 1, we find "=^' " = 4' "' = Go- Substituting ia {u + v + w)y = l, we obtain y = 2i; and therefore a; =45, z=10. 292. Here S, is the coefficient of a"" in the expansion of (l + a){l + ax) ... {1 + ax^-^). Thus (l + a){l + ax}...{l + ax''-^) = l + Sj^ii + S3a'+...+S^'-+.... Write ax for a, then {l + ax)(l + ax^)... (l + aa;") = l + Siaa! + S2aV+...+S'ra'j;'-+...; .-. (l + ax"){l + Sia + S„a2+... + S'^a''+...} = (l + a){l + fi'iax + S2a2j;2+... + fi'Xj;''+...}. 45" y "24" 2 "lo' «a; t)y wz y ""i"" ^T' 368 MISCELLANEOUS EXAMPLES. [PAGE Equate coefficients of a"-'' ; thus .-. (1 - »»-•) S„_,= (1 - x^^) a;"-'-! S„-r-i- Write r + 1 for r, then (1 _ a;"--!) s:„_^i = (1 _ :c'^!=) ^"-'•-^ S„_^ ; (1 - ^"-'■-2) S„_,_,, = (1 - aj-'+s) x"---' S„-,_3 ; (1 - a:'^!) S^+i = (1 - a;"-"-) a:' S,. Multiply these results together; then, since the product of the binomial factors ia the same on each side, we have 293. If a, 6, c are not integers, we can find an integer m which will make ma, mb, mc integral. The expression I 1 + J ( 1 + — y— J ( 1 + - ) is the product of ma + mb + mc positive factors, since the sum of any two of the quantities «., 6, c is greater than the third. The arithmetic mean of these factors is {ma + b-c) + {mb + c-a) + {mc + a-b) ma + mb + mc and is therefore equal to unity. Hence the ahove expression is less than iwnHmi+mc^ or unity. [Art. 253.] 294:. (1) The given expression = (262c2 + 2c^a^ + 2a262 _ a< - 6^ _ c*) {a^ + b^ + c^) - 8a-6V ^ a" (6" + c") + b* (c^ + o2) + c^ (a2 + b^) - «« - ftS - c6 - 2a-bV = {b- + c^~a'){c^ + a^-b'){a^ + b^-c-']. 523.] MISCELLANEOUS EXAMPLES. 369 (2) We have {x + y + z)*+ {- x + y + z)*=2 {x* + 6x^ {y + zf + {y + z)*} ; and (x-y + z)* + {x + y-z)* = 2{x* + Gx''{y-zf + {y-z)''}. .: {x +y + zy+ {-X + y + z)* + (x -y + z)* + {x + y - z)* = 4:{x* + y^ + ^ + 62/2^2 + 6a2a;2+ 6xh/). By putting x=p + y, y=y + a, z=a + p, and dividing throughout by 4, we obtain the required result. 295. The required sum is equal to the coefficient of a;' in the product of the series 1+ x+ x^ + ... + x''+..., l + 2x + 2V + ... + 2'-a:'-+..., l + 3x + 3''x^+ ... + 3'-x^+ ..., and therefore to the coefficient of x'' in the expression 111 1 1-x' l-2x'l-3a; 1-nx 1 _A_ B C ® (l-x)(l-2x)(l-8x)...(l-?7j;)~l^"x ■^l-2x"'"l-3a;" (- 1)"-' then by the theory of Partial Fractions, we find -4 = ^ ^ ; (_ 1 )n-2 2»-i _ (- 1)"-° (re -1)2"-' (-l)»-'3»-i , ,^„ ,(re-l)(n-2)3"-i , ^ =— rm — 3— =(- 1 ^ — iir-\ — r ; and so on. Hence the sum required is equal to the coificieut of x*" in (_l)n-i ^ 1 (n-l)2''-i (re-l)(n-2)3"-i _J \ n-l (1-x l-2x "'' |2 'l-bx whence the result easily follows. 296. The giyen expression is equal to ,_3„|,_^3^(3re^4H3re^,_ J_ Now 1, ^j — - , , ... are respectively the coefficients of l.J l.J.o (1 — x)~- (l — x]"^ x'"-'-, x'""*, x^"-', ... in the expansion of (l-x)~', ^ — —— , - — ^-'— , ... H. A. K. 2i 370 MISCELLANEOUS EXAMPLES. [PAGE , 3n-3 (3n-4)f3m-5) 1 X^ .T* = the coefficient of x^"~' in : ' 1-x 2{l-x)^ ■d(l-xf^ ■■■ 1 / x'^ \ 1 1 + a:^ This series is the expansion of -^ log ( 1 + = ) , or -^ log ^ . X \ X — Xy X X — X I 1 + a^} Therefore the required aeries = l — 37i jthe coefficient of x^" in logr — -^[ . , 1 + a;' , a:6 x' , ,,^, x^" Now log- r,=x3--+^-... + (-l)J>-i — + ... / , X* X6 X^P \ If n is odd, the coefficient of x"" is (- l)"~i - , or - ; n n 12 1 if n is even, the coefficient of x^" is (- 1)" ' - + n- > or - ;t- . Thus the value of the required series is l-3ng),orl-3„(-l), according as n is odd or even. These expressions are equal to -2, +2 respectively. Thus in each ease the series is equal to 2 (- 1)". 297. We have 2a - X = — , so that X = 2a : u u 6^ 6' ^3 similarly « = 2a ; z = 2a ; « = 2a . z y ^ X „ „ b^ b^ 62 62 Hence x = 2a-T; ;; r, — — • 2a— 2a— 2a— x As in Art. 438, or as in XXXI. a. Ex. 1, we have The successive convergents to the continued fraction are 2a ia'-b^ 8a" - 4a62 16a* - 120=62 + 6* T' 2a ' iw'-V ' 8a3 - 4a62 ' (16a* - 12a262 + 6*) x - 6' (80^ - iab^) (8a3-4a62)x-62(4a2-62) 524.] MISCELLANEOUS EXAMPLES. 371 Equating this last expression to x and simplifying, we have ia (2a2 _ 62) x^ - Sa^ {2a' - 6') x + iaV' {2a' -b') = 0; or ia (2a2 - 6^) {x'' - 2ax + b"-) = 0. Hence unless 2a'-' -6^=0, we have x' — 2ax + b''=0. Similar equations hold for y, z, V,, and therefore x = y = z = u. If however 2a^-b''=0 the above equation is satisfied; in this case we have 2a2 2a^ 2a' _ -4a* '"" " 2a- 2a- y ~2aV^4^' that is, X (2a -y) = 2a'', which is the remaining equation ; hence the given equations are not independent. 298. From the third equation, z= ; hence substituting in the x + y first two equations, we have ax{x + y)=c{y + l), by {x + y)=c {x + 1). From the first of these equations, we find ax' -c ... c (x - 1) y = , so that x + y= — '- . c-ax '^ c-ax On substitution in the second of the above equations, we obtain fc(ax2-c)(x-l)-(x + l)(c-ax)2=0; or {ab - a') x' + Px" + Qx + {be - c''] say. For the discussion of the roots it is immaterial whether a is greater or less than 6; let us suppose that a is the greater. There are however two cases to consider, namely when oa, and when c<:a. Let us tabulate the signs of the expression b {ax' - c) (x - 1) - (x + 1) (c - ax)' for different values of x. (1) Suppose oa, so that - > ./ - >1, When X = - 00 , the sign is the same as that of a^ — aft, and therefore is + ; when x= - 1, the sign is + ; when x=l, the sign is - ; when ^ = A / - 1 the sign ia - ; when x = - , the sign is + ; when x= + 00 , the sign is - . Hence there are three changes of sign, and therefore three real roots. [If a < 6, the expression is negative when x= — oo and positive when X = + 00 , and there are still three changes of sign.] MISCELLANEOUS EXAMPLES. 372 ■' ■ (2) Suppose c < a, so that 1 > » / - > - ; then [page when x= - oo , the sign is + ; when x= -1, the sign is - ; when x= when x = - , the sign is + ; \/ a' the sign is + ; when x = l, the sign is - ; when a: = + 00 , the sign is - . Hence as before there are three real roots. The product of the roots is -; . or —+: . ab — a'a{b-a) Similarly we can shew that y has three real values, and by interchanging a and 6, we see that the product of these values is -H r^; hence the b(a-b) second part of the question follows at once. Since the values of x and y are real the values of z must be real. 299. Denote the expression on the left by X; then A' = A F E F B V EDO ax-by - cz bx + ay az + cx bx + ay -ax + by -cz bz + cy az + cx bz + cy -ax + by -cz f both sides by 1 a b c = bz-cy. X y z Then X{bz-cy) = ax-by - cz bx + ay az + ex a;(a= + 62 + c2) ^ (a-! + 62 + ^2) z(a2+62 + c2) a(x^+y'> + z') b(x^ + y^ + z^) c(x'' + y^ + z'>) = (a'> + 6= + c2)(a;2 + j/2 + 02) ax -by- cz bx + ay az + ex it y z a b c 524.] MISCELLANEOUS EXAMPLES. ' 373 Multiply the second row by - a, the third by - x and add to the first ; then the last determinant - ax - by - cz ;<; y z a be hence {bz-cy)X= {a^ + b^ + c'} (x' + y^ + z"^) {ax + bTj + cz) [bz - cij) ; whence the result follows. 300. Suppose that at first he walked x miles a day and worked y hours a day, and that the investigation lasted n days. On the r"" day he walked x + r-1 miles and worked y + r-1 hours, and therefore counted - {x + r-1) {y + r-1) words ; m being some constant. Hence - {xy + {x + l){y + l) + (x + 2j{y + 2)+ ... to « terms} = 232000; that is, nxj/ + (x + 2/)(l+2 + 3 + ...+ »"^l) + {12 + 2-4-32 + .. . + {n-l)2} = 232000m; nin — 1) 1 or nxy + — ^-- — ' (■x + y) + ^n{n-l) (2n - 1) = 232000m. But --xy = 12000 ; that is,xy = 12000m. mi. r n{n-l)/l 1\ 1 , ,,,„ ,,1 116 ,,, Therefore n+ ^ „ (- + -] + ■^n{n-l){2n-l) — = -^ (1 . 2 \x yj 6 ^ ' xy G At the end of half the time he had counted 62000 words ; therefore by changing n into - , we have n n{n-2)fl , 1 \x yj 2i " ' xy b 2 8 Multiply this equation by 2 and subtract the result from (1), then n^n-l) 1_- rf /I r 4 \x y 4 xy On the last day he counted 72000 words ; therefore - {x + n-l){y + n-l) = 72000 ; m that is, xy + {n-l)mj + {n-'Vf' = 72000m ; .-. l + (7i-l)fi+-V(»-l)''— =6; '\x yj ^ ' xy ' 374 MISCELLANEOUS EXAMPLES. [PAGE 524. or {n-l)(~ + -)+(n-l)''~ = 5. Thus Ti- + -+ f=9; and (n-1) -^- + - + Y=j; 9 ■■4(«-l) 5' .-. 5re2-36>i + 36 = 0, or (5n - 6) (k - G) = 0. Substituting m=6, we find from (1) and (2), , /I IN 55 40 ^n 1\ 5 13 15- + -) + — = —; 3- + -) + — = — ; \x yj xy S \x yj xy 6 1,17 1 1 whence - + -=—, _ = _; X y 12 xy 12 that is, x + y = 7, xy = 12; or x = 3, y = i. CAJBBfilDGB : PEINTED BY C. J. CLAY, il.A. AND 80K3, AT THE UNITHaSITY PRESS By the same Authors. ELEMENTARY ALGEBRA FOR SCHOOLS. Fourth Edition. Eevised and enlarged. Globe 8vo. (bound in maroon coloured cloth), 38. 6d. With Answers (bound in green coloured cloth), in. 6d. PEEFACE TO THE FOURTH EDITION. In the Second Edition a chapter on the Theory of Quadratic Equations was introduced ; the book is now further enlarged by the addition of chapters on Permutations and Combinations, the Binomial Theorem, Logarithms, and Scales of Notation. These have been abridged from our Higher Algebra, to which readers are referred for a fuller and more exhaustive treatment. The very favourable reception accorded to the first three editions leads us to hope that in its present more complete form the work will be found suitable as a first text-book for every class of student, and amply suf&cient for all whose study of Algebra does not extend beyond the Binomial Theorem. The Schoolmaster says : — " Has so many points of excellence, as compared with its prede- cessors, that no apology is needed for its issue... The plan always adopted by every good teacher of frequently recapitulating and making additions at every recapitulation, is well carried out." The Educational Times says : — "... A very good book. The explanations are concise and clear, and the examples both numerous and well chosen." Nature says : — "This is, in our opinion, the beat Elementary Algebra for school use. ..We con- fidentiv recommend it to mathematical teachers, who, we feel sure, will find it the best book of its kind for teaching purposes." The Acailemy says : — "Buy or borrow the book for yourselves and judge, or write a better... A higher te.xt-b6ok is on its way, Tliis occupies sufficient ground for the generality of boys." HIGHER ALGEBRA. A Sequel to "Elementary Algebra for Schools." Third Edition, revised. Crown 8vo. 7s. 6d. The AthenePiim says:— "The Elementary Algebra b^ the same authors, which has already reached a third edition, is a work of such exceptional merit that those acquainted with it will form high expectations of the sequel to it now issued. Nor will they be disappointed. Of the autliors' Higher Al;/cbra, as of their Elementary Algebra, we unhesitatingly assert that it is by far the best work of its kmd with which we are acquainted. It supplies a want much felt by teachers." The Academy says:— "Is as admirably adapted for College students as its predecessor was for schools. It is a well-arranRed and well-reasoned-out treatise, and contains much that we have not met with before in similar works. For instance, we note as specially good the articles on Convergency and Divergency of Series, on the treatment of Series generally, and the treatment of Continued Fractions... The book is almost indispensable and will be found to improve upon acquaintance." The Saturday Review says:— "They have presented such difficult parts of the subject as Convergency and Divergency of Series, Series generally, and Probability with great clearness and fulness of detail.. .No student preparing for the University should omit to get this work in addition to any other he may have, for he need not fear to hnd here a mere repetition of the old story. We have found much matter of interest and many valuable hints.., We would specially note the examples, of which there are enough, and more than enough, to try any student's powers." MACMILLAN AND CO., LONDON. By the same Authors. ALGEBRAICAL EXERCISES AND EXAMINATION PAPERS. To accompany "Elementary Algebra." Third Edition. Globe 8vo. 2s. M. The Schoolmaster says:— "As useful a collection of examination papers in algebra as we ever met with. . . .Each 'exercise' is calculated to occupy about an hnur in its solution, lie- sides these, there are 35 examination papers set at various competitive examinations durinc the last three years. The answers are at the end of the book. We can strongly recommend the volume to teachers seekiuR a well-arranged series of tests in algebra." The Educational T'nnes says:— "It is only a few months since we spoke in high praise of an Elementary Algebra by the same authors of the above papers. We can speak also in high praise about this little book. It consists of over a hundred progressive miscellaneous exercises, followed by a collection of papers set at recent examinations. The exercises are timed, as a rule, to take an hour. . . . Messrs Hall and Knigh* have had plenty of experience, and have put that experience to good use " The Spectator says :— "The papers are arranged for about an hour's work, and will be found a useful addition to the school text-book." The Irish Teachers' Journal says:— "We know of no better work to place in the hands of junior teachers, monitors, and senior pupils. Any person who works carefully and steadily through this book could not possibly fail in an examination of Elementary Algebra. . . . We cou- gratulate the authors on the skill displayed in the selections of examples." ARITHMETICAL EXERCISES AND EXAMINATION PAPERS. With an Appendix containing Questions in LOGARITHMS AND MENSURATION.^ With Answers. Second Edition. Globe 8vo. 2s. 6d. The Schoolmaster says: — "An excellent hook to put into the hands of an upper class, or for use by pupil teachers. It covers the whole ground of arithmetic, and has an appendix con- taining numerous and well-selected questions in logarithms and mensuration. In addition to these good features there is a collection of fifty papers set at various public examinations during the last few years." The (Cambridge Review says:— "All the mathematical work these gentlemen have pven to the public is of genuine worth, and these exercises are no exception to the rule. The addition of the logarithm and mensuration questions adds greatly to the value.'* The Educational Times says:— "The questions have been selected from a great variety of sources: London University Matriculation; Oxford Locals— Junior and Senior; Cambridge Locals — .Junior and Senior ; Army Preliminary Examinations, etc. As a preparation for examina- tion the book will be found of the utmost value." The School Board Chronicle says :— "The work cannot fail to be of immense utility." A TEXT BOOK OF EUCLID'S ELEMENTS, including Alternative Proofs, together with additional Theorems and Exercises, classified and arranged. By H. S. Hall, M.A., and F. H. Stevens, M,A., Masters of the Military and Engineering Side, Clifton College. IJookL 1*. I Books L— IV. 3*. I Books T.—VL U.6d. Uooks I. and IL U 6d. | Books in.— VL 3*. | Book XL [In the press. The Cambridge Review says:— "To teachers and students alike we can heartily recom- mend this little edition of Euclid's Elements. The proofs of Euclid are with very few exceptions retained, but the unnecessarily complicated expression is avoided, and the steps of the proofs are so arranged as readily to catch the eye. Prop. 10, Hook IV., is a good example of how a long proposition ought to be written out. The candidate for mathematical honours will find intro- duced in their proper places short sketches of such subjects as the Pedal Line, Maxima and Minima, Harmonic Division, Concurrent Lines, &c, quite enough of each for all ordinary require- ments. Useful notes and easy examples are scattered throughout each book, and set's of hard examples are given at tlie end. The whole is so evidently the work of practical teachers, that we feel sure it must soon displace every other Euclid." The Journal of Education says:- "The most complete introduction to Plane Geometry based ou Euclid's Elements thiit we have yet seen." The Practical Teacher says:— ''One of the moat attractive books on Geometry that has yet fallen into our hands." The Literary World says: — "A distinct advance on all previous editions." The Irish Teachers' Journal says : — " It must rank as one of the very best editions of Euclid in the language." MACMILLAN AND CO., LONDON. Cornell University APR 2 5 2007 Mathematics Library m